Abelian and Dihedral equiangular tight frames of redundancy $2$

This paper studies group frames ($G$-frames) where the unitary group representation can be projective. When the group is abelian, for most combinations $N, n$, we show that $ETF(N,n)$ can only exist for genuinely projective group representations. In particular, cyclic-group frames for such parameters do not exist. We also give a characterization of all dihedral tight frames and dihedral $ETF(2n,n)$, using which, we conclude that regular dihedral $ETF(2n,n)$ must be genuinely projective. Following that, we give a characterization of regular dihedral $ETF(2n,n)$ in terms of certain structured skew Hadamard matrices. We then show that Paley $ETF(2n,n)$ and its doubling are both of this type. Finally, we classify all regular dihedral $ETF(2n,n)$ for $n\le 22$ up to switching equivalence.