Let $G$ be a countable monoid and let $A$ be an Artinian group (resp. an Artinian module). Let $Σ\subset A^G$ be a closed subshift which is also a subgroup (resp. a submodule) of $A^G$. Suppose that $Γ$ is a finitely generated monoid consisting of pairwise commuting cellular automata $Σ\to Σ$ that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the valuation action of $Γ$ on $Σ$ satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissible group subshifts.