Groups in which the centralizer of every involution is solvable were classified by D. Gorenstein and various co-authors. Also J. G. Thompson classified finite groups such that the normalizer of every solvable subgroup is solvable. Results of this kind were in some ways more general than the problem you asked about.

Groups in which the centralizer of every involution is solvable were classified by D. Gorenstein and various co-authors. Also J. G. Thompson classified finite groups such that the normalizer of every non-identity solvable subgroup is solvable. Results of this kind were in some ways more general than the problem you asked about.

Groups in which the centralizer of every involution is solvable were classified by D. Groenstein and various co-authors. Also J.G. Thompson classified finite groups such that the normalizer of every solvable subgroup is solvable. Results of this kind were in some ways more general than the problem you asked about.

Groups in which the centralizer of every involution is solvable were classified by D. Gorenstein and various co-authors. Also J. G. Thompson classified finite groups such that the normalizer of every solvable subgroup is solvable. Results of this kind were in some ways more general than the problem you asked about.