Unpublished results of S Straus and W Browder state that two notions of homotopy equivalence for manifolds with smooth group actions—isovariant and equivariant—often coincide under a condition called the Gap Hypothesis; the proofs use deep results in geometric topology. This paper analyzes the difference between the two types of maps from a homotopy theoretic viewpoint more generally for degree one maps if the manifolds satisfy the Gap Hypothesis, and it gives a more homotopy theoretic proof of the Straus–Browder result.
"Isovariant mappings of degree 1 and the Gap Hypothesis." Algebr. Geom. Topol. 6 (2) 739 - 762, 2006. https://doi.org/10.2140/agt.2006.6.739