We introduce the notion of impulse extension equations for linear fixed-time impulsive differential equations (IDEs) with strictly inhomogeneous impulses. These differential equations can be thought of as representing the underlying processes for which such linear fixed-time IDEs are a limiting case. We will establish basic existence and uniqueness results, and then develop sufficient conditions for existence and uniqueness of periodic solutions, and study an example of an IDE whose every solution is periodic, even though almost all impulse extension equations which resembles it admit no periodic or bounded solutions.