The canonical answer to this question is of course the celebrated solution by Hill, Hopkins and Ravenel to the Kervaire invariant one problem

> Hill, Michael A., Michael J. Hopkins, and Douglas C. Ravenel. ["On the nonexistence of elements of Kervaire invariant one."](https://annals.math.princeton.edu/2016/184-1/p01) *Annals of Mathematics* (2016): 1-262 ([arXiv:0908.3724](https://arxiv.org/abs/0908.3724)).

Beside that amazing paper, equivariant homotopy theory can be used in the computation of Picard groups of some local categories. For example the following paper uses $C_4$-equivariant homotopy theory to study the Picard group of the $K(2)$-local category

> Beaudry, Agnes, Irina Bobkova, Michael Hill, and Vesna Stojanoska. "Invertible $ K (2) $-Local $ E $-Modules in $ C_4 $-Spectra." ([arXiv:1901.02109](https://arxiv.org/abs/1901.02109)) (2019).

In both cases the intuition is that equivariant techniques are useful to run descent arguments along a Galois extension (typically by some small subgroup of the Morava stabilizer group). For example the slice spectral sequence allows one to resolve Borel $G$-spectra (i.e. "spectra with a $G$-action") by pieces that are not Borel anymore but which are "simpler" in some sense.