Good involutions of twisted conjugation subquandles and Alexander quandles

We completely describe good involutions of free quandles and subquandles of twisted conjugation quandles of groups, including all Alexander quandles. As an application, we enumerate good involutions of linear quandles, and we provide explicit mappings for those up to order 23 via a computer search. Along the way, we completely characterize connected, involutory Alexander quandles, which may be of independent interest.