More detail on Jones's 30 manifold and analogous constructions can be found in my <a href="http://math.wayne.edu/~rrb/papers/manif.pdf"> Extended powers of manifolds and the Adams spectral sequence </a>, Cont. Math. 271, 41--51.  The basic idea in it dates from 1979 or so (immediately upon seeing Jones's construction) but didn't make it into print for another 20 years.  There are also comments about the homotopy theoretic approach to the problem in <a href="http://math.wayne.edu/~rrb/papers/fin_conj_handout.pdf">  the talk I gave in Edinburgh </a> in 2011 at the workshop on the Kervaire invariant. 

The main point is that there is a small set of homology classes in low dimension (dimension 2 for Jones' construction) which one wants to realize as the fundamental class of a manifold with tangent bundle realized by a permutation representation of $\pi_1$.  To get the 30 dimensional Kervaire manifold from $S^7$ one only needs a two manifold and an appropriate representation in $S_4$.  ($D_8$ will do:  only the Sylow 2-subgroup matters.)  To get from $S^7$ to the 62 dimensional Kervaire manifold requires a 6-manifold with a representation of $\pi_1$ in $S_8$.  Unfortunately, no such manifold exists.  In dimension 126, one would need a 14 manifold, and presumably such also fails to exist. So, one needs to find another approach that relaxes the input data needed.