Subdifferentials (limiting Fréchet, Clarke) of the composition \(f\circ \lambda \) of extended real valued permutation invariant functions \(f\) and the eigenvalue vector function \(\lambda \) of a symmetric matrix \(X\) are calculated using the transformation to principles axes. If \(U\) is an orthogonal matrix, such that \(\text{Diag}(\lambda (X)) =UXU^{T}\) is the diagonal transform of \(X\) and \(f(Px) =f(x) \) for each permutation matrix \(P\) then under assumptions usual in nonsmooth analysis for \(f\) it is shown \(\partial (f\circ \lambda) (X) =\{ U^{T}\text{Diag}(\mu) U\mid U\) as above, \(\mu \in\partial f(x)|_{x=\lambda (X)}\}\). The unique regular subdifferentiability, Fréchet-differentiability and strict differentiability of \(f\) is equivalent to those one of \(f\circ \lambda \). Further, the invariance under Young subgroups is investigated. As an example the \(k\)th eigenvalue function \(\lambda _{k}(X) \) and the associated \(k\)th order statistic is considered. The main proofs, also for some used differential geometric aspects of spaces of real matrices, are given.