Are $C^\infty$ exotic spheres $C^k$ exotic? The only theory of exotic spheres that I know is of $C^\infty$ structures on them; that is, that there are plenty of spheres (in dimensions $n \geq 7$ that are homeomorphic but not diffeomorphic. To keep this question reasonable, I'll restrict to the $n=7$ case. Are the exotic $S^7$s diffeomorphic for any $C^k$ with $k \geq 1$? Has there been any serious study about the structure of (the cobordism group of) $C^k$ exotic spheres for finite positive $k$? If so, what are some references?

Any $C^k$ structure on a manifold for $k > 0$ can be uniquely promoted (modulo diffeomorphism) to a $C^\infty$ structure; furthermore, the corresponding map between $C^k$ structures modulo equivalence to $C^\infty$ structures modulo equivalence is bijective. See this MathOverflow question, for example.