If X is a dense subspace of a space B, then B is called an extension of X, and the subspace Y = B∖X is called a remainder of X. We study below, how the properties of remainders of spaces influence the properties of these spaces. In particular, we establish the following fact: if Y is a remainder of a topological group G in an extension B of G, and every closed pseudocompact G_δ-subspace of Y is compact, and B contains a nonempty compact subset Φ of countable character in B such that G∩Φ≠∅, then G is a paracompact p-space (Theorem 2.3). This fact plays a key role in the proofs of the similar statements for images and preimages of topological groups under perfect mappings (see Theorems 3.1, 3.2 and 3.4).