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 is compact connected and abelian, its endomorphisms extensible along morphisms into compact connected groups also include (in addition to the obvious trivial endomorphism and the identity). Connectedness cannot be dropped on either side in this last statement.