We prove extension theorems for several geometric properties such as asymptotic property C (APC), finite decomposition complexity (FDC), strict finite decomposition complexity (sFDC) which are weakenings of Gromov's finite asymptotic dimension (FAD).The context of all theorems is a finitely generated group G with a word metric and a coarse quasi-action on a metric space X. We assume that the quasi-stabilizers have a property P 1 , and X has the same or sometimes a weaker property P 2 . Then G also has property P 2 .We show some sample applications, discuss constraints to further generalizations, and illustrate the flexibility that the weak quasi-action assumption allows.

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