We study the effects of information discrepancy across sub-populations on their ability to simultaneously improve their features in strategic learning settings. Specifically, we consider a game where a principal deploys a decision rule in an attempt to optimize the whole population's welfare, and agents strategically adapt to it to receive better scores. Inspired by real-life settings, such as loan approvals and college admissions, we remove the typical assumption made in the strategic learning literature that the decision rule is fully known to the agents, and focus on settings where it is inaccessible. In their lack of knowledge, individuals try to infer this rule by learning from their peers (e.g., friends and acquaintances who previously applied for a loan), naturally forming groups in the population, each with possibly different type and level of information about the decision rule. In our equilibrium analysis, we show that the principal's decision rule optimizing the welfare across subgroups} may cause a surprising negative externality; the true quality of some of the subgroups can actually deteriorate. On the positive side, we show that in many natural cases, optimal improvement is guaranteed simultaneously for all subgroups in equilibrium. We also characterize the disparity in improvements across subgroups via a measure of their informational overlap. Finally, we complement our theoretical analysis with experiments on real-world datasets.