EDT0L solutions to equations in group extensions

We show that the class of groups where EDT0L languages can be used to describe solution sets systems equations is closed under direct products, wreath products with finite groups, and passing index subgroups. also add contain a product hyperbolic as subgroup list solutions expressed an language. This includes dihedral Artin groups. rational constraints in virtually abelian have solutions, addition recognisable contraints any system preserves property having solutions. These are respect quasigeodesic normal forms. discuss space complexity which for these constructed.