Let U be a unipotent group of Lie type over a finite field F_q. Let pi:U->U/U^1 be the natural projection. Assume, for simplicity, that U^2 = {e}. (Here U^1 = [U,U], U^2 = [U,[U,U]], etc.) Say I have a subgroup V of U/U^1.

Is there a subgroup H < U such that

(a) pi(H) = V,

(b) H\cap U^1 = <[H,H]> ?

(Notice that H\cap U^1 must always contain <[H,H]>, which depends only on V.)