In Top, how do conjugate homorphisms of groups induce homotopies of classifying spaces? They exist— there's an abstract proof, but how is BG → BH written in terms of the homotopy??
If G and H are only H-spaces, a homomorphism induces a map of a projective plane. In Top, how do conjugate H-maps induce homotopies of maps of projective planes??
If I interpret your first question correctly, it's equivalent to asking for a group $G$ and $g \in G$ how to see the map $f_g\colon BG \rightarrow BG$ induced by the homomorphism $G \rightarrow G$ taking $x$ to $g x g^{-1}$.
Let $\ast \in BG$ be the base point, which I'll assume is non-degenerate (so the inclusion $\ast \hookrightarrow BG$ is a cofibration). The map $f_g$ should take $\ast$ to $\ast$. Let $\gamma\colon [0,1] \rightarrow BG$ be the loop based at $\ast$ corresponding to $g$. Using the homotopy extension property, we can find a homotopy $\phi_t\colon BG \rightarrow BG$ with $\phi_0 = \text{id}$ and $\phi_t(\ast) = \gamma(t)$. The desired $f_g$ is then $f_g = \phi_1$. This map is homotopic to the identity, but of course the base point moves during the homotopy.
There's a construction of $BG$ used quite often by geometric-oriented homotopy theorists that makes the homotopies quite explicit. In this model for $BG$ you start by "resolving" $G$ to the homotopy-equivalent space of finite subsets of $[0,1]$ labelled by elements of $G$, with the rules that if two labelled points collide then their labels multiply. To make this definition precise you have to take the disjoint union $\bigsqcup_n \Delta_n \times G^n$ and mod out by the collision relation, given by maps of the boundary facets to $\Delta_{n-1} \times G^{n-1}$. Here $\Delta_n = \{ (t_1,\dotsc, t_n) : 0 \leq t_1 \leq \dotsb \leq t_n \leq 1 \}$ is the $n$-simplex. You similarly allow for the deletion of points labelled by the neutral element.
In this model there are two actions of $G$ on this "resolved" $G$, one by inserting a point on the left, and another by inserting a point on the right. You mod out by one of these actions and you have a model for $EG$, and mod out by both to get $BG$.
Anyhow, so say you have a homomorphism $f : G \to H$, then the induced map of classifying spaces $Bf$ simply takes a collection of $G$-labelled points in $[0,1]$ and replaced them by the $H$-labelled points by applying $f$ to the labels. So if all the labels are conjugated $x^{-1} f(g) x$ by a common element you can slide the leftmost $x^{-1}$ off the $0$ end of the interval, the rightmost $x$ off the $1$ end of the interval, and all the other $x^{-1}$ and $x$ pairs can slide together and cancel. That would be the homotopy.
Dev Sinha gives the appropriate images to model this in his presentation: Geometry of Eilenberg-MacLane Spaces II - Iterated Bar.
It appears around the 18-minute mark. But he goes into detail in the case of $\mathbb RP^\infty$. Which might be what you are looking for.
