I haven't checked but here are some guidelines to show the group is Hopfian.
Write the original group (given by 4x4 triangular matrices) as G=D\ltimes U with D=Z^2 and U its unipotent part. Set U^2=[U,U] and U^3=[U,U^2], which is central and naturally isomorphic to Fp[t,1/t]. Our group is H=G/M, where M\subset U^3 is generated by F_p[t] and t^{-2}. Let f be a surjective endomorphism of H.
check that the center of G is precisely U^3. It follows that f induces a surjective endomorphism of G/U^3. Since this group is linear, it is Hopfian so this is an automorphism of G/U^3.
Describe the group of automorphisms of G/U^2 = Z^2\ltimes F_p[t,1/t]^3. (it should be reasonably easy to describe).
Deduce a description of the group of automorphisms of G/U^3, or at least describe how these automorphisms act on U^2/U^3, showing that modulo something, the coefficient 12 is multiplied by a monomial w.t^a (w\in F_p*) and the coefficient 24 is multiplied by vt^b. So, taking a commutator (that should kill the "modulo something"), we obtain that in the action of f on H, the coefficient 14 should be multiplied by a monomial. This multiplication should stabilize M so this is multiplication by a scalar in F_p*, which implies that f is actually an automorphism.