As an important fundamental task of numerous real-world applications such as social network analysis and online advertising/marketing, several prior works studied influential community search, which retrieves a community with high structural cohesiveness and maximum influences on other users in social networks. However, previous works usually considered the influences of the community on arbitrary users in social networks, rather than specific groups (e.g., customer groups, or senior communities). Inspired by this, we propose a novel Top-M Reverse Influential Community Search (TopM-RICS) problem, which obtains a seed community with the maximum influence on a user-specified target community, satisfying both structural and keyword constraints. To efficiently tackle the TopM-RICS problem, we design effective pruning strategies to filter out false alarms of candidate seed communities, and propose an effective index mechanism to facilitate the community retrieval. We also formulate and tackle a TopM-RICS variant, named Top-M Relaxed Reverse Influential Community Search} (TopM-R2ICS), which returns top-M subgraphs with relaxed structural constraints and having the maximum influence on a user-specified target community. Comprehensive experiments have been conducted to verify the efficiency and effectiveness of our TopM-RICS and TopM-R2ICS approaches on both real-world and synthetic social networks under various parameter settings.