This paper presents a complete solution to the nonstandard H2 output feedback control problem for continuous descriptor systems where unstable and nonproper weighting functions are used. In such a problem, the desired controller has to satisfy two conditions simultaneously: (i) the closed-loop is admissible and has a minimum H2 norm, (ii) only the internal stability of a part of the closed-loop is sought. The condition of the existence of such a controller is deduced. An explicit characterization of the optimal solution is also formulated, based on two generalized algebraic Riccati equations (GAREs) and two generalized Sylvester equations. A numerical example is included to illustrate the validity of the proposed results.