Let $X$ be a reasonable topological space (say one that has the homotopy type of a finite CW complex) and consider a subset $\Gamma$ of $X \times X$ so that the projection $p:\Gamma \to X$ onto the first factor is surjective (making no assumptions on the other projection $q$):

$$ X \stackrel{p}{\gets} \Gamma \stackrel{q}{\to} X$$

Let's call a function $f:X \to X$ supported by $\Gamma$ if $(x,f(x)) \in \Gamma$ for each $x$ in $X$. In other words, the graph of $f(x)$ must be a subset of $\Gamma$. Here's my question:

Is there a "classification" of all possible endomorphisms $H_\bullet f: H_\bullet X \to H_\bullet X$ on (integral or field) homology induced by continuous $f$'s which are supported by (a fixed) $\Gamma$?

As an initial example, note that if the first projection $p$ has contractible point-inverses, then there is a unique endomorphism supported by $\Gamma$.

Let $X$ be a reasonable topological space (say one that has the homotopy type of a finite CW complex) and consider a subset $\Gamma$ of $X \times X$ so that the projection $p:\Gamma \to X$ onto the first factor is surjective (making no assumptions on the other projection $q$):

$$ X \stackrel{p}{\gets} \Gamma \stackrel{q}{\to} X$$

Let's call a function $f:X \to X$ supported by $\Gamma$ if $(x,f(x)) \in \Gamma$ for each $x$ in $X$. In other words, the graph of $f(x)$ must be a subset of $\Gamma$. Here's my question:

Is there a "classification" of all possible endomorphisms $H_\bullet f: H_\bullet X \to H_\bullet X$ on (integral or field) homology induced by continuous $f$'s which are supported by (a fixed) $\Gamma$?

As an initial example, note that if the first projection $p$ has contractible point-inverses, then there is a unique endomorphism supported by $\Gamma$. I'd also be happy with answers to the "opposite" question

Is there a (homological?) obstruction to a given endomorphism $\phi: HX \to HX$ being induced by a $\Gamma$-supported $f$?

Thanks!