Fixed point free automorphisms of groups related to finite fields

Let G=Fq⋊〈β〉 be the semidirect product of the additive group of the field of q=pn elements and the cyclic group of order d generated by the invertible linear transformation β defined by multiplication by a power of a primitive root of Fq. We find an arithmetic condition on d so that every endomorphism of G is determined by its values on (1,1) and (0,β). When that is the case, we determine the fixed point free automorphisms of G. If d equals the odd part of q−1 then we count the fixed point free automorphisms of G—such exist only when p is a Fermat prime.