In multigroup data settings with small within-group sample sizes, standard F-tests of group-specific linear hypotheses can have low power, particularly if the within-group sample sizes are not large relative to the number of explanatory variables. To remedy this situation, an alternative test statistic is presented that shares information across groups. Each group-specific test has potentially much larger power than the standard test, while still exactly maintaining a target type-I error rate if the hypothesis for the group is true. The proposed test for a given group uses a statistic that has optimal marginal power under a prior distribution derived from the data of the other groups.