Let Λ be a finite dimensional algebra with an action by a finite group G and A : = Λ * G the skew group algebra. One of our main results asserts that the canonical restriction-induction adjoint pair induced by the skew group algebra extension Λ ⊂ A induces a poset isomorphism between the poset of G -stable support τ -tilting modules over Λ and that of ( mod G )-stable support τ -tilting modules over A . We also establish a similar poset isomorphism between posets of appropriate classes of silting complexes over Λ and A . These two results generalize and unify the preceding results by Zhang–Huang, Breaz–Marcus–Modoi and the second and the third authors. Moreover, we give a practical condition under which τ -tilting finiteness and silting discreteness of Λ are inherited by A . As applications we study τ -tilting theory and silting theory of the (generalized) preprojective algebras and the folded mesh algebras. Among other things, we determine the posets of support τ -tilting modules and of silting complexes over preprojective algebra Π ( 𝕃 n ) of type 𝕃 n .