We investigate when a group of the form  $G\times \mathbb {Z}^m\ (m\geq 1)$ has the finitely generated fixed subgroup property of automorphisms (
$G\times \mathbb {Z}^m\ (m\geq 1)$ has the finitely generated fixed subgroup property of automorphisms ( $\mathrm {FGFP_a}$), by using the BNS-invariant, and provide some partial answers and nontrivial examples.
$\mathrm {FGFP_a}$), by using the BNS-invariant, and provide some partial answers and nontrivial examples.