Isomorphism types of definable (maximal) cofinitary groups

Kastermans proved that consistently ${⨁}_{{\mathrm{\aleph }}_1}{\mathbb{Z}}_2$ has a cofinitary representation. We present a short proof that ${⨁}_{\mathfrak{c}}{\mathbb{Z}}_2$ always has an arithmetic cofinitary representation. Further, for every finite group $F$ we construct an arithmetic maximal cofinitary group of isomorphism type $({\ast }_{\mathfrak{c}}\mathbb{Z})\times F$. This answers an implicit question by Schrittesser and Mejak whether one may construct definable maximal cofinitary groups not decomposing into free products.