On preservation of automatic continuity

A group G is called automatically continuous if any homomorphism from a completely metrizable or locally compact Hausdorff group to G has open kernel. In this paper, we study preservation of automatic continuity under group-theoretic constructions, focusing mainly on groups of size less than continuum. In particular, we consider group extensions and graph products. As a consequence, we establish automatic continuity of virtually poly-free groups, and hence of non-exceptional spherical Artin groups. On the other hand, we show that if G is automatically continuous, then so is any finitely generated residually G group, hence, for instance, all finitely generated residually free groups are automatically continuous.