Let $\Theta_n$ be the set of orientation-preserving diffeomorphism classes of homotopy spheres, with abelian group structure given by #. Then for any smooth manifold $M^n$ one defines the "inertia subgroup" as

$I(M)=${$\Sigma\in\Theta_n$ | $M$#$\Sigma\cong M$}

In other words, the inertia group is the set of homotopy spheres which fix the smooth structure on $M$.

Does anyone know an example of a diffeomorphism (as explicit as possible) between an $M$ and $M$#$\Sigma$, where $\Sigma$ is as exotic sphere? Moreover, if the dimension were sufficiently large can we always find a diffeomorphism which induces the identity on $\pi_1$?

(The big question I'm trying to think about is "If $\Sigma$ is in the inertia group of $M$, how does its smooth structure get 'unraveled' in $M$?")