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Since Horadam and de Launey introduced the cocyclic framework on combinatorial designs in the 1990s, it has revealed itself as a powerful technique for looking for (cocyclic) Hadamard matrices. Ten years later, the series of papers by Kotsireas, Koukouvinos and Seberry about Hadamard matrices with one or two circulant cores introduced a different structured approach to the Hadamard conjecture. This paper is built on both strengths, so that Hadamard matrices with cocyclic cores are introduced and studied. They are proved to strictly include usual Hadamard matrices with one and two circulant cores, and therefore provide a wiser uniform approach to a structured Hadamard conjecture. [ABSTRACT FROM AUTHOR]