  In this paper, we study the hereditary properties of module
$(\phi,\varphi)$-amenability on Banach algebras. We also define the concept of
module character contractibility for Banach algebras and obtain
characterizations of module character contractible Banach algebras in terms of
the existence of module $(\phi,\varphi)$-diagonals. We introduce module
approximately character amenable Banach algebras. Finally, for every inverse
semigroup $S$ with subsemigroup $E$ of idempotents, we find necessary and
sufficient conditions for the $\ell^1(S)$ and its second dual to be module
approximate character amenable (as a $\ell^1(E)$-module).
