Given a covering space $p \colon X \to Y$, we get an injection $p^\* \colon \pi\_1(X) \to \pi\_1(Y)$, and we know that the image $p^\*(\pi\_1(X))$ is normal in $\pi\_1(Y)$ if an only if $p$ is *regular*, that is if the deck group of the covering $p$ acts transitively on preimages of points in $Y$. This is a satisfyingly topological criterion for a group-theoretic property.


Is a similarly topological criterion possible for the case where $p^\*(\pi\_1(X))$ is not just normal in $\pi\_1(Y)$ but characteristic?


Edit: If it helps, $X$ can be assumed to be aspherical. The initial context in which this came up was surfaces.



When do covering spaces correspond to characteristic subgroups?
Suppose $X$ is a $K(\pi,1)$. Then by definition every endomorphism, hence every automorphism $f\_\star\colon \pi\_1(X,x) \to \pi\_1(X,x)$ may be realized by a continuous map $f\colon (X,x) \to (X,x)$.


If $p\colon (\tilde X,\tilde x) \to (X,x)$ is a covering map such that $p\_\star(\pi\_1(\tilde X, \tilde x))$ is characteristic in $\pi\_1(X,x)$, then every homotopy equivalence $f\colon (X,x) \to (X,x)$ preserves the image $p\_\star(\pi\_1(\tilde X, \tilde x))$ so by the lifting criterion (see Hatcher chapter 1, for example), there exists a lift $\tilde f\colon (\tilde X,\tilde x) \to (\tilde X,\tilde x)$ such that $p\tilde f = fp$. In fact, if every homotopy equivalence lifts to $\tilde X$, then the converse of my use of the lifting criterion implies that $p\_\star(\pi\_1(\tilde X,\tilde x))$ is characteristic in $\pi\_1(X,x)$.


