In this note we first show, using results of McDuff, that for a Cr-manifold S, diffeomorphic to the interior of a compact manifold with boundary, the class of all Cr-diffeomorphisms lying in a one parameter group of S generates the connected component of 1S in Diffr (S, S). Then we use this result to obtain two extension theorems for stratified maps defined on some strata of a stratified space [Xscr]. Our extension theorems hold for Mather's abstract stratified sets, for Whitney (b)-regular, Bekka (c)-regular, Verdier (w)-regular and Lipschitz-regular stratified spaces.