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Let $(\Gamma,+,F)$ be a finitely generated $\mathbb Z[F]$-module where $F$ isan injective endomorphism of the abelian group $\Gamma$. We restrict ourselvesto a finite automa presentable subclass, introduced by J. Bell and R. Moosa in"F-sets and finite automata. J. Th\'eor. Nombres Bordeaux 31 (2019), no. 1,101-130" and define an expansion containing the $\mathcal F$-sets defined by R.Moosa and T. Scanlon in "Am. J. Math. 126 (2004), no. 3, p. 473-522", whereevery automatic subset is definable.

A decidable expansion of $(\Gamma,+,F)$ with the independence property