Let $G$ be a discrete group and let $(X,x_0)$ be a based Eilenberg-MacLane space for $G$, so there is a fixed isomorphism $\pi_1(X,x_0) = G$ and the universal cover $\widetilde{X}$ is contractible. The homology of $X$ is thus the same as the homology of $G$.

Choose some $\gamma \in G$, and let $c_{\gamma} \in \text{Aut}(G)$ be the inner automorphism that conjugates elements of $G$ by $\gamma$. It is standard that $c_{\gamma}$ acts as the identity on $H_k(G)$ for all $k$.

Since $(X,x_0)$ is an Eilenberg-MacLane space for $G$, there is a based map $\psi_{\gamma}\colon (X,x_0) \rightarrow (X,x_0)$ that induces $c_{\gamma}$ on $\pi_1(X,x_0)$, and in fact $\psi_{\gamma}$ is unique up to based homotopy.

**Question**: Is it possible to write down $\psi_{\gamma}$ in some natural way where in particular it is obvious that it acts as the identity on $H_k(X)$? All the proofs I know that inner automorphisms act trivially on homology are either very algebraic or are based on very specific models of Eilenberg--MacLane spaces that mimic the algebraic constructions, and I'd like to see this fact topologically/geometrically.