I am trying to understand a proof I found in a paper by Wagoner about delooping of algebraic K-theory (proposition 1.2 for those interested). For this I have a fibration $BE\to BG\to BG^{ab}$, with $E=[G,G]$, and I want to understand the action of $BG^{ab}$ on $H_*(BE)$. The paper seems to assume that this action is given by conjugation, in which case by a lemma from the paper (1.3) and some standard group homology, the action is trivial in the case of interest.
What I don't understand, is why the action of $G^{ab}$ on $H_*(E)$ coming from the homotopy fibration would be the conjugation action coming from group homology.
Any help is appreciated, thank you.