When Miss Scarlet & The Duke premiered, it seemed to be following the stereotypical format of any period piece "lady detective" series. The character arrives and forthwith finds herself in need of solving a crime. Despite lots of people saying "A lady detective!?" in shock and awe, she does the job with aplomb, winning the supporting cast's hearts and minds, and opens up shop.
The second episode hinted the series won't go easy on her, putting her in the dock several times, talking openly about money woes, and so on. But the triumph of her second case and Moses's addition suggested this was just the "ups and downs" episode that regularly informs Episode 2 in many mystery series.
Margaret: You work for men. You do whatever they ask. And you take their money and hope they use you again. You may as well be working in a brothel.
So it was a bit of a surprise when Episode 3 begins with Eliza still sitting in an empty office, struggling to get clients to walk in the door. Worse, it turns out the client is Duke, and he doesn't even give her the respect to turn up sober or on time. Nor does he think twice about the job he's asking her to do. The assignment is to infiltrate a suffragette group. Their leader, Margaret Fairfax (Caitlin Drabble), is a bit of a firebrand and wants keeping an eye on.
Eliza agrees because she needs the money, but she's already torn at the idea of spying on her fellow women, who are, after all, fighting for the same thing she is. She's also terrible at maintaining her cover, rapid-fire spitting her cover story to Flora Mountford (Geraldine McAlinden) in the most improbable way. Lucky for her, PC Honeychurch turns up to break things up, and afraid he'll ruin her cover, Eliza uses him to show herself a woman of the cause and slaps his face, getting herself arrested and winning Fairfax's trust.
But the whole thing is short-lived. At the second meeting, Margaret is gone, and Duke shows up looking for her, announcing she's wanted for murder, blowing Eliza's cover entirely in the process. But even though he's ruined the job and ruined Eliza's reputation among these women, he demands she keeps working the case to help him find Fairfax, refusing to pay her unless she agrees.
From here on out, the plot is immaterial. Margaret did it, admitting as much when she tracks down Eliza at her office. The real axis that drives this week is that Eliza has a real problem on her hands. No one respects her. Not Ivy, who thinks this whole suffragette thing is nonsense, and so are women who demand rights. Not Duke, who is using her in the ugliest way to get himself ahead in the eyes of his bosses without letting them know he's not doing all the work. Duke's minions don't respect her, Honeychurch openly laughs at her. Even the women in the movement don't respect her. When brought in for questioning, Flora sneers at Eliza; she only wants to talk to a real detective. Eliza manages to threaten Flora into submission, but the internalized misogyny is all around her.
The only characters whose respect she has won so far are Rupert and Moses, both of whom themselves are subjected to bigotry, though Rupert at least has money shielding him from the worst of it. But neither is exactly a sound support system, nor can they step in when Eliza needs backing to get people to listen to her. (Duke practically attempts to drive Moses off.) In other period pieces with women detectives (Miss Fisher, The Alienist), there's always a white man of good breeding around who vouches for our heroine and with their respect can automatically create a space in which she can work to earn the respect of those around her. Eliza doesn't have that. She's truly alone.
Worse, there's a level where the show doesn't let us forget Eliza has undermined herself. Every "I feel faint, I need some water" she pulls, every time she gets herself into deeper trouble by failing to keep her mouth shut, it's eroding ground. Every time Eliza does anything reinforcing the stereotype of the flighty woman (even when it benefits her in the moment), it's slowly backfiring on her overall trajectory. Moreover, even if she does attempt to be "more emotionless like a man" (which the show also makes sure we understand is utter nonsense), it's not getting her anywhere.
Perhaps most importantly, it's not who she is. She's young; she's excitable, she's nervy. (Notably, most successful female detectives are older and more self-assured, like in Unforgotten or Prime Suspect.) Those traits are why she succeeds. In this case, despite Duke screwing everything up for her right and left, all while claiming he's trying to help her, she puts together Fairfax is a real danger, more than Duke realized when he assigned the case. She's not just going to protest Parliament and their laws; she's going to blow them sky-high.
Eliza stops the bombing plot, though it's Duke who gets to drag Margaret in for confession. There, she charges Eliza to be honest with herself; her detective business isn't blazing a trail for anyone but herself. Once again, Duke is there with words, but not deeds. Despite him claiming that Margaret's claims are baseless, he then goes into his boss, Superintendent Stirling (Nick Dunning). He uses Eliza's work, yet again, to make himself look better and help him angle for a promotion.
Eliza at least realizes there is something she can do to help push the cause of women forward, or at least one woman: Ivy. Reminded at the top of the episode Ivy never learned to read or write, Eliza takes it upon herself to fix that and give a woman the keys to a larger world. It's not solving crimes; it's not getting ahead or even respect. It's not solving her problems with Duke or her inability to be independent of him. But it's at least one change she can make in the world, one difference, and that's good enough for today.