  We investigate charge-parity (CP) and non-CP outer automorphism of groups and
the transformation behavior of group representations under them. We identify
situations where composite and elementary states that transform in exactly the
same representation of the group, transform differently under outer
automorphisms. This can be instrumental in discriminating composite from
elementary states solely by their quantum numbers with respect to the outer
automorphism, i.e. without the need for explicit short distance scattering
experiments. We discuss under what conditions such a distinction is
unequivocally possible. We cleanly separate the case of symmetry constrained
(representation) spaces from the case of multiple copies of identical
representations in flavor space, and identify conditions under which
non-trivial transformation in flavor space can be enforced for composite
states. Next to composite product states, we also discuss composite states in
non-product representations. Comprehensive examples are given based on the
finite groups $\Sigma(72)$ and $D_8$. The discussion also applies to
$\mathrm{SU}(N)$ and we scrutinize recent claims in the literature that
$\mathrm{SU}(2N)$ outer automorphism with antisymmetric matrices correspond to
distinct outer automorphisms. We show that outer automorphism transformations
with antisymmetric matrices are related by an inner automorphism to the
standard $\mathbb{Z}_2$ outer automorphism of $\mathrm{SU}(N)$. As a direct
implication, no non-trivial transformation behavior can arise for composite
product states under the outer automorphism of $\mathrm{SU}(N)$.
