Let H be a connected reductive group over a non-archimedean local field k and let F ⊂ Autk(H) be a finite group of order not divisible by p, the residual characteristic of k. Let G = (H F )◦ be the identity component of the subgroup of H consisting of points fixed by F. The main theorem of this paper asserts that the Bruhat-Tits building B(G) of G can be identified with the set of F-fixed points of B(H). Several special cases of this theorem have been known previously. When E/k is a finite totally ramified Galois extension, H = ResE/k G, and F = Gal(E/k), the condition p #F is simply that E/k is tamely ramified. In this case, our main theorem is a well-known but unpublished theorem of G. Rousseau. Recently, one of us (G.P.) found a simple proof of this theorem [P]. When G is a classical group, realized in the standard way as the identity component of the group of fixed points of an involution φ of a general linear group H , Bruhat and Tits [BT4] have given a description of B(G) as a subset of B(H). In particular, it follows from their description that B(G) = B(H) when p = 2. Recently, this fact has been rediscovered by J. Kim and A. Moy [KM], and independently, a simple proof has been given by [GY]. Let H be the split form of Spin(8). Then one can choose an outer automorphism φ ∈ Aut(H) of order 3 such that G = H is of type G2. From a joint work [GY] by W.T. Gan and one of us (J.Y.), it is known that B(G) = B(H) is true for arbitrary p.