Consider a pair \((G,A)\), where \(G\) is a locally compact Abelian group, \(H\) is a discrete subgroup of \(G\) such that the quotient group \(G/H\) is compact and \(A\) is an automorphism of \(G\) such that \(A(H)\) is a proper subgroup of \(H\). Denote \(L^2(G,\mu)\), where \(\mu\) is the Haar measure on \(G\), by \(L^2(G)\). The author considers the problem of finding the orthogonal wavelets in \(L^2(G)\). He indicates a scheme for solving this problem and constructing the orthogonal wavelets \(\psi_i\) in \(L^2(G)\), where \(i=1,2,\dots, s\) and \(s= \text{card}(H/A(H))\). Two theorems are established with necessary conditions in terms of the pair \((G,A)\) for the possibility to apply the above mentioned scheme.