  Let $A$ be a Banach algebra and $X$ be a compact Hausdorff space. Given
homomorphisms $ \sigma \in Hom(A)$ and $\tau \in Hom(C(X, A))$, we introduce
induced homomorphisms $\tilde{\sigma}\in Hom(C(X, A)) $ and $\tilde{\tau}\in
Hom(A)$, respectively. We study when $\tau$-(weak) amenability of $C(X, A)$
implies $\tilde{\tau}$-(weak) amenability of $A$. We also investigate where $
\sigma$-weak amenability of $A$ yields $\tilde{\sigma}$-weak amenability of
$C(X, A)$.
