This paper provides solvability conditions for state synchronization with homogeneous discrete-time multi-agent systems (MAS) with a directed and weighted communication network under full-state coupling. We assume only a lower bound for the second eigenvalue of the Laplacian matrices associated with the communication network is known. For the rest the weighted, directed graph is completely arbitrary. Our solvability conditions reveal that the synchronization problem is solvable for any nonzero lower bound if and only if the agents are at most weakly unstable (i.e., agents have all eigenvalues in the closed unit disc). However for a given lower bound, we can achieve synchronization for a class of unstable agents. We provide protocol design for at most weakly unstable agents based on either a direct eigenstructure assignment method or a standard H₂ discrete-time algebraic Riccati equation (DARE). We also provide a protocol design for strictly unstable agents based on the standard H₂ DARE.