Genetic engineering combined with CRISPR technology has developed to the point that gene drives can, in theory, be engineered to cause extinction in countless species. Success of extinction programs now rests on the possibility of resistance evolution, which is largely unknown. For CRISPR technology, resistance may take many forms, from mutations in the nuclease target sequence to specific types of non-random population structures that limit the drive. We develop mathematical models of various deviations from random mating to consider escapes from extinction-causing gene drives. We use a version of Maynard Smith's haystack model to show that population structure can enable drive-free subpopulations to be maintained against gene drives. Our main emphasis, however, is sib mating in the face of recessive-lethal and Y-chromosome drives. Sib mating easily evolves in response to both kinds of gene drives and maintains mean fitness above 0, with equilibrium fitness depending on the level of inbreeding depression. Environmental determination of sib mating (as might stem from population density crashes) can also maintain mean fitness above 0. Translation of mean fitness into population size depends on ecological details, so understanding mean fitness evolution and dynamics is merely the first step in predicting extinction. Nonetheless, these results point to possible escapes from gene-drive-mediated extinctions that lie beyond the control of genome engineering.