Summary: In this paper, we consider Einstein gravity in the presence of a class of nonlinear electrodynamics, called power Maxwell invariant (PMI). We take into account \((2 + 1)\)-dimensional spacetime in Einstein-PMI gravity and obtain its black hole solutions. Then, we regard pure \(F(R)\) gravity as well as \(F(R)\)-conformally invariant Maxwell (CIM) theory to obtain exact solutions of the field equations with black hole interpretation. Finally, we investigate the conserved and thermodynamic quantities and discuss about the first law of thermodynamics for the mentioned gravitational models.