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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 $\Theta^n$

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There is no natural Lie group structure on a connected sum of a Lie group and an exotic sphere. Where would it possibly come from?

Also, the implication you are trying to derive is wrong. Note that $S^3$ is a Lie group and hence so is a product of several $S^3$s. I'm not an expert on surgery theory but it's well-known that the inertia group of a product of spheres is trivial which means that the opposite of the statement you are after holds: the connected sum of a product of spheres with an exotic sphere is never orientably diffeomorphic to the original manifold. So for example the inertia group of $S^3\times S^3\times S^3$ is trivial while $\Theta_9$ has order 8.

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