The authors study Schur algebras with a projective basis, i.e. central simple \(k\)-algebras which are isomorphic to a twisted group algebra over \(k\). Their main results are as follows: 1. If the twisted group algebra \(k^\alpha G\) is a division algebra with centre \(k\), then the commutator subgroup of \(G\) is cyclic. Moreover 2. \(G\) is nilpotent and \(k^\alpha G\) can be expressed as a tensor product corresponding to the Sylow subgroups of \(G\). This gives a reduction to \(p\)-groups. Now 3. If \(G\) is a \(p\)-group and \(D=k^\alpha G\) is a division algebra with centre \(k\), then (i) \(D\) is a tensor product of cyclic algebras (with projective bases), where all but possibly one are symbol algebras, provided that either \(p\) is odd or \(p=2\) and \(\sqrt{-1}\in k\). If (ii) \(p=2\) and \(\sqrt{-1}\notin k\), then for \(D=k^\alpha G\), where \(G\) is a 2-group, \(D\) is a tensor product of quaternion algebras, or \(\text{char }k=0\) and \(D\) is a tensor product of quaternion algebras and a crossed product of a certain Abelian 2-group with a projective basis. The proof proceeds by a detailed analysis and uses the following Factorization Lemma. Let \(k^\alpha G\) be a twisted group division algebra, where \(|G |\) is invertible in \(k\). Given a normal subgroup \(H\) of \(G\) such that \(k^\alpha H\) is \(k\)-central, one has \(k^\alpha G\cong k^\alpha H\otimes_k k^\beta E\), where \(k^\beta E\) is a \(k\)-central twisted group algebra with finite group \(E\) and \(\beta\in H^2(E,k^*)\).