Let $f:G\to H$ be a morphism of topological groups and let
$H^{hG}:=Map_G(EG, H)$
be the homotopy centralizer. Here $H$ is considered a $G$-space via the map $f$, that is, $g\cdot h=f(g)hf(g^{-1})$, and the object $H^{hG}$ retains the structure of a topological group by pointwise multiplication.
Is it generally true that the classifying space $B(H^{hG})$ of $H^{hG}$ satisfies
$B(H^{hG})\simeq Map(BG, BH)_{Bf}$?
Here the right-hand side is the connected component corresponding to $f$ in the (unbased) mapping space between the classifying spaces of $G$ and $H$.
I have seen special cases of this statement in the literature, for example [Dwyer-Zabrodski, "Maps between classifying spaces", Theorem 1.1]. But is the general statement above also true?