We exhibit an \(\textsf {AC}^{3}\) isomorphism test for Fitting-free groups given by their Cayley tables, a class for which isomorphism testing was previously known to be in \(\textsf{P}\) (Babai, Codenotti, & Qiao; ICALP ’12). In sharp contrast, we show that for permutation groups, isomorphism testing of Fitting-free groups is at least as hard as Graph Isomorphism and Linear Code Equivalence (the latter being GI-hard and having no known subexponential-time algorithm). Lastly, we show that any Fitting-free group of order n is identified by \(\textsf {FO}\) formulas (without counting) using only \(O(\log \log n)\) variables. This is in contrast to the fact that there are infinite families of Abelian groups that are not identified by FO formulas with \(o(\log n)\) variables (Grochow & Levet, FCT ’23).