Extensible Endomorphisms of Compact Groups

We show that the endomorphisms of a compact connected group that extend to endomorphisms of every compact overgroup are precisely the trivial one and the inner automorphisms; this is an analogue, for compact connected groups, of results due to Schupp and Pettet on discrete groups (plain or finite). A somewhat more surprising result is that if $$\mathbb {A}$$ is compact connected and abelian, its endomorphisms extensible along morphisms into compact connected groups also include $$-\textrm{id}$$ (in addition to the obvious trivial endomorphism and the identity). Connectedness cannot be dropped on either side in this last statement.