A characterization of the reversibility of linear cellular automata

Let $G$ be a group, let $\mathbb{K}$ be a field, and let $V$ be a $\mathbb{K}$-vector space. We prove that there exists a bijective linear cellular automaton $V^G\to V^G$ whose inverse is not a cellular automaton if and only if $G$ is not locally finite and ${\dim }_{\mathbb{K}}V\ge \mathrm{\mid }\mathbb{K}{\mathrm{\mid }}^{{\mathrm{\aleph }}_0}$. This answers an open problem proposed by T. Ceccherini-Silberstein and M. Coornaert.