An extended derivation (endomorphism) of a (restricted) Lie algebra
L
L
is an assignment of a derivation (respectively) of
L
′
L’
for any (restricted) Lie morphism
f
:
L
→
L
′
f:L\to L’
, functorial in
f
f
in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of
L
′
L’
to every
f
f
; and (b) if
L
L
is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then
L
L
is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman.
In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms.