Frobenius structures are omnipresent in arithmetic geometry. We show that over suitable rings, Frobenius endomorphisms define differential structures and vice-versa. This in turn can be used to construct differential modules in characteristic p and p-adic differential equations with "nice" Galois groups. Moreover, this construction leads to p-adic and t-adic Galois representations of the Galois group of k(t) having images like the Dikson group G_2 (here k is the algebraic closure of F_p).