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." *Annals of Mathematics* (2016): 1-262 (arXiv: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) (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.