$\DeclareMathOperator\Map{Map}\newcommand{\B}{\mathrm{B}}\newcommand{\h}{\mathrm{h}}$Let $f:G\to H$ be a morphism of topological groups and let

$H^{\h G}:=\Map_G(\mathrm{E}G, H)$

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 $H^{\h G}$ retains the structure of a topological group by pointwise multiplication.

Is it generally true that the classifying space $\B(H^{\h G})$ of $H^{\h G}$ satisfies

$\B(H^{\h G})\simeq \Map(\B G, \B H)_{\B f}$?

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?