  We introduce the notion of a graded group action on a graded algebra or,
which is the same, a group action by graded pseudoautomorphisms. An algebra
with such an action is a natural generalization of an algebra with a super- or
a pseudoinvolution. We study groups of graded pseudoautomorphisms, show that
the Jacobson radical of a group graded finite dimensional associative algebra
$A$ over a field of characteristic $0$ is stable under graded
pseudoautomorphisms, prove the invariant version of the Wedderburn-Artin
Theorem and the analog of Amitsur's conjecture for the codimension growth of
graded polynomial $G$-identities in such algebras $A$ with a graded action of a
group $G$.
