  Let $G$ and $\tilde G$ be reductive groups over a local field $F$. Let $\eta
: \tilde G \to G$ be a $F$-homomorphism with commutative kernel and commutative
cokernel. We investigate the pullbacks of irreducible admissible
$G$-representations $\pi$ along $\eta$. Following Borel, Adler--Korman and Xu,
we pose a conjecture on the decomposition of the pullback $\eta^* \pi$. It is
formulated in terms of enhanced Langlands parameters and includes
multiplicities. This can be regarded as a functoriality property of the local
Langlands correspondence. We prove this conjecture for three classes: principal
series representation of split groups (over non-archimedean local fields),
unipotent representations (also with $F$ non-archimedean) and inner twists of
$GL_n, SL_n, PGL_n$. Our main techniques involve Hecke algebras associated to
Langlands parameters. We also prove a version of the pullback/functoriality
conjecture for those.
