  We provide combinatorial interpretation for the $\gamma$-coefficients of the
basic Eulerian polynomials that enumerate permutations by the excedance
statistic and the major index as well as the corresponding
$\gamma$-coefficients for derangements. Our results refine the classical
$\gamma$-positivity results for the Eulerian polynomials and the derangement
polynomials. The main tools are Br\"and\'en's modified Foata--Strehl action on
permutations and the recent triple statistic (des, rix,aid) equidistibuted with
(exc, fix, maj).
