  We describe homomorphisms $\varphi:H\rightarrow G$ for which the codomain is
acylindrically hyperbolic and the domain is a topological group which is either
completely metrizable or locally countably compact Hausdorff. It is shown that,
in a certain sense, either the image of $\varphi$ is small or $\varphi$ is
almost continuous. We also describe homomorphisms from the Hawaiian earring
group to $G$ as above. We prove a more precise result for homomorphisms
$\varphi:H\rightarrow {\rm Mod}(\Sigma)$, where $H$ as above and ${\rm
Mod}(\Sigma)$ is the mapping class group of a connected compact surface
$\Sigma$. In this case there exists an open normal subgroup $V\leqslant H$ such
that $\varphi(V)$ is finite. We also prove the analogous statement for
homomorphisms $\varphi:H\rightarrow {\rm Out}(G)$, where $G$ is a one-ended
hyperbolic group. Some automatic continuity results for relatively hyperbolic
groups and fundamental groups of graphs of groups are also deduced. As a
by-product, we prove that the Hawaiian earring group is acylindrically
hyperbolic, but does not admit any universal acylindrical action on a
hyperbolic space.
