The class of strongly exotic topological groups (SEG) introduced by W. Banaszczyk [Additive subgroups of topological vector spaces. Berlin: Lect. Notes Math. 1466 (1991; Zbl 0743.46002)] is studied in the paper. By definition, a topological group \(G\) is called SEG if there is no non- trivial weakly continuous representation \(T : G \to B(H)\) in a Hilbert space \(H\), where \(B(H)\) denotes the \(B^*\)-algebra of bounded linear operators on \(H\). The author considers the behaviour of the class of SEGs with respect to the operations of inductive and projective limits and proves that SEG is a full subcategory of the category of topological groups. A method of the construction of non-metrizable SEGs is presented.