Summary: We show in this paper that for a non-compact LCA group \((G, \tau), {\tau}\) has exactly \(2^{2^{k(G)}}\) predecessors in \(\mathcal{G}_2(G)\). This answers a problem posed in the literature affirmatively. Denote by \(\mathfrak{N}\) the class of all LCA groups \((G, \tau)\) such that \(P_2(\tau)\) is precompact. It is shown that for an LCA group \(G, G \in \mathfrak{N}\) if and only if \(G \cong \mathbb{R}^n \times H\), where \(n\) is a non-negative integer and \(H\) is an LCA group with an open compact subgroup \(N\) such that \(H / N \in \mathfrak{M}\). As an application of this result, we extend a well known result on discrete abelian groups to the case of LCA groups. We show that if \((G, \tau)\) is a non-compact LCA group and \({\sigma}\) is a predecessor of \({\tau}\) in \(\mathcal{G}_2(G)\), then the connected component of \((G, \tau)\) coincides with the connected component of \((G, \sigma)\). It is also shown that for a non-compact LCA group \((G, \tau)\), if \({\sigma}\) is a predecessor of \({\tau}\) in \(\mathcal{G}_2(G)\), then the equality \(i b(G, \tau) = i b(G, \sigma)\) holds. This partially answer a question posed in the literature.