Let ${\mathbb{M}}={\mathbb{N}}^D$ be the positive orthant of a $D$-dimensional lattice and let $({\mathcal{G}},+)$ be a finite abelian group. Let ${\mathfrak{G}} \subseteq{\mathcal{G}}^{\mathbb{M}}$ be a subgroup shift, and let $\mu$ be a Markov random field whose support is ${\mathfrak{G}}$. Let $\Phi: {\mathfrak{G}} \longrightarrow {\mathfrak{G}}$ be a linear cellular automaton. Under broad conditions on ${\mathcal{G}}$, we show that the Cesaro average $N^{-1} \sum_{n=0}^{N-1} \Phi^n (\mu)$ converges to a measure of maximal entropy for the shift action on ${\mathfrak{G}}$.