pvec = {0.03, 0.06, 0.09, 0.12, 0.15};
Xpvec = Table[0, 5]; n = 1;
While[n < 6,
fr = FindRoot[
1 - 3/(8 Xp^3) (2 Xp^2 - 1 + (1 + 2 Xp) Exp[-2 Xp]) ==
pvec[[n]], {Xp, 0.01, 0.5}];
xtemp = Xp /. fr; Xpvec[[n]] = xtemp; n++];
Xpvec
gives
{0.0408762, 0.0836014, 0.128334, 0.175253, 0.22456}