Amenability-Like properties of C(X,A)

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 $C(X, A)$.