In [3] a sufficient condition for the input-output finite-time stability (IO-FTS) of impulsive systems has been provided in terms of a feasibility problem involving a coupled difference/differential LMI (D/DLMI). In this paper we show that such condition is also necessary; moreover an alternative necessary and sufficient condition for IO-FTS is proved. The latter condition requires the solution of a coupled differ-ence/differential Lyapunov equation (D/DLE) and is shown to be more efficient from the computational point of view than the D/DLMI based condition. In order to prove the main result, we exploit the definition of controllability Gramian extended to impulsive systems. An example illustrates the benefits of the proposed technique.