Virtual endomorphisms of the group $pg$

A virtual endomorphism of a group G is a homomorphism of the form φ:H→ G, where $H<G$ is a subgroup of finite index. A virtual endomorphism φ:H→ G is called simple if there are no nontrivial normal φ-invariant subgroups, that is, the φ-core is trivial. We describe all virtual endomorphisms of the plane group $pg$, also known as the fundamental group of the Klein bottle. We determine which of these virtual endomorphisms are simple, and apply these results to the self-similar actions of the group. We prove that the group $pg$ admits a transitive self-similar (as well as finite-state) action of degree d if and only if d≥ 2 is not an odd prime, and admits a self-replicating action of degree d if and only if d≥ 6 is not a prime or a power of $2$.