Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial higher homotopy groups). Then $\pi\_1(X)$ acts on the universal cover of $X$ via deck transformations, and on the universal cover of $Y$ via 


$$([\sigma],y)\mapsto [\sigma \circ f]y$$


If g is another continuous map whose lift to the universal covers is equivariant under these actions, is g homotopic to f? 



Are two equivariant maps between aspherical topological spaces homotopic?
In the language of equivariant homotopy theory, your question is as follows. You have a group homomorphism $\phi\colon G\to H$, which you use to make $EH$ into a $G$-space, and then you ask whether $[EG,EH]^G$ is a singleton. In fact, it is true more generally that $[EG,Y]^G$ is a singleton whenever $Y$ is a $G$-space that is nonequivariantly contractible. The reason is as follows. We can write $EG$ as the colimit of a sequence $X\_k$, where $X\_k$ is obtained from $X\_{k-1}$ by attaching $G\times B^{n+1}$ along $G\times S^n$ for some $n$. Thus, to extend an equivariant map $X\_{k-1}\to Y$ equivariantly over $X\_k$ is the same as to extend a nonequivariant map $S^n\to Y$ nonequivariantly over $B^{n-1}$, and this can be done in an essentially unique way because $Y$ is assumed to be nonequivariantly contractible. (This kind of argument is called "obstruction theory".)


