A continuous aperture array (CAPA)-based multi-group multicast communication system is investigated. An integral-based CAPA multi-group multicast beamforming design is formulated for the maximization of the system energy efficiency (EE), subject to a minimum multicast SE constraint of each user group and a total transmit power constraint. To address this non-econvex fractional programming problem, the Dinkelbach's method is employed. Within the Dinkelbach's framework, the non-convex group-wise multicast spectral efficiency (SE) constraint is first equivalently transformed into a tractable form with auxiliary variables. Then, an efficient block coordinate descent (BCD)-based algorithm is developed to solve the reformulated problem. The CAPA beamforming design subproblem can be optimally solved via the Lagrangian dual method and the calculus of variations (CoV) theory. It reveals that the optimal CAPA beamformer should be a combination of all the groups' user channels. To further reduce the computational complexity, a low-complexity zero-forcing (ZF)-based approach is proposed. The closed-form ZF CAPA beamformer is derived using each group's most representative user channel to mitigate the inter-group interference while ensuring the intra-group multicast performance. Then, the beamforming design subproblem in the BCD-based algorithm becomes a convex power allocation subproblem, which can be efficiently solved. Numerical results demonstrate that 1) the CAPA can significantly improve the EE compared to conventional spatially discrete arrays (SPDAs); 2) due to the enhanced spatial resolutions, increasing the aperture size of CAPA is not always beneficial for EE enhancement in multicast scenarios; and 3) wider user distributions of each group cause a significant EE degradation of CAPA compared to SPDA.