<p>This paper considers the inverse problem of determining the time-dependent coefficient in the fractional wave equation with the generalized Riemann–Liouville (Hilfer) time derivative. In this case, the direct problem is the initial value problem for this equation with Cauchy type nonlocal initial conditions. To represent the solution of the direct problem, the fundamental solution of this equation is constructed and properties of this solution are investigated. The fundamental solution contains Fox’s H-functions widely used in fractional calculus. Using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient, which will be used in investigation of the inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation, the contracted mapping principle is applied. The local existence and global uniqueness results and the conditional stability estimate of the solution are proven</p>