Twisted reversibility in SL (2, ℤ) and PSL (2, ℤ)

Given a group automorphism φ:G→G, one has an action of G on itself by φ-twisted conjugacy, given by g⋅x=gxφ(g−1). An element x∈G is called φ-twisted reversible if it is φ-twisted conjugate to its inverse. To analyze this notion, we develop the framework of extended stabilizers associated with reversible elements and study their structural features. We also address reversibility when G acts on arbitrary groups, enabling the study of reversing symmetry beyond the classical conjugation action with particular emphasis on automorphic reversibility. In this paper, we classify the subgroups of SL(2,Z) and PSL(2,Z) that stabilize elements under twisted conjugacy actions and identify the possible cardinalities of the extended stabilizers. In addition, we characterize the sets of twisted reversers and the subgroups that φ-stabilize elements of the infinite dihedral group D∞.