A note on the finitely generated fixed subgroup property

We study when a group of form $G\times {\mathbb{Z}}^m(m\ge 1)$ has the finitely generated fixed subgroup property of automorphisms (${\mathrm{F}\mathrm{G}\mathrm{F}\mathrm{P}}_a$), by using the BNS-invariant, and provide some partial answers and non-trivial examples.