Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

We describe homomorphisms $φ:H\rightarrow G$ 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 $φ$ is small or $φ$ is almost