For finite groups, isovariant maps that strictly preserve isotropy groups satisfy a Blakers-Massey theorem and a Freudenthal suspension theorem, once suspension is defined using a complete G-universe as a homotopy terminal object.
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An isovariant Blakers--Massey theorem
For finite groups, isovariant maps that strictly preserve isotropy groups satisfy a Blakers-Massey theorem and a Freudenthal suspension theorem, once suspension is defined using a complete G-universe as a homotopy terminal object.