  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 $-\mathrm{id}$ (in addition to the obvious
trivial endomorphism and the identity). Connectedness cannot be dropped on
either side in this last statement.
