See Smooth manifold with non-trivial inertia group? (wrt homotopy spheres) for the definition of $\Theta_n$ and inertia subgroups.
I'm wondering what can be said about Lie groups. If $M^n$ is an n-dimensional manifold with Lie group structure and $\Sigma^n$ is a homotopy n-sphere, is there a Lie group structure on $M\#\Sigma$ that is in some sense compatible with the original structure? If this new group structure is isomorphic to the old structure, this implies Lie group isomorphism, correct?
What I would like to see is that there is a canonically induced Lie group structure on $M\#\Sigma$ and that this structure is isomorphic to that of $M$, and hence the inertia group for Lie groups is the full $\Theta^n$