Restricting diffeomorphisms of a closed n-manifold M to an embedded disc D^n induces a fibration sequence Diff(M,D^n) -> Diff(M) -> Emb(D^n,M), where Emb(D^n,M) is the space of embeddings of the disc and Diff(M,D^n) the group of diffeomorphisms fixing it. Connect summing M with an exotic sphere S does not affect the homotopy type of the two outer terms in the sequence, so one might expect it to be hard to detect the possible exotic nature of M#S by means of homotopical properties of its group of diffeomorphisms. Combining computations in stable homotopy theory with recent advances in manifold theory by Galatius and Randal-Williams, I will present results on the behavior of the cohomology of BDiff(M) when taking the connected sum with an exotic sphere.