Picard groups of quasi-Frobenius algebras and a question on Frobenius strongly graded algebras

Our initial aim was to answer the question: does the Frobenius (symmetric) property transfers from a strongly graded algebra to its homogeneous component of trivial degree? Related to it, we investigate invertible bimodules and the Picard group of a finite dimensional quasi-Frobenius algebra \(R\). We compute the Picard group, the automorphism group and the group of outer automorphisms of a \(9\)-dimensional quasi-Frobenius algebra which is not Frobenius, constructed by Nakayama. Using these results and a semitrivial extension construction, we give an example of a symmetric strongly graded algebra whose trivial homogeneous component is not even Frobenius. We investigate associativity of isomorphisms \(R^*\ot_RR^*\simeq R\) for quasi-Frobenius algebras \(R\), and we determine the order of the class of the invertible bimodule \(H^*\) in the Picard group of a finite dimensional Hopf algebra \(H\). As an application, we construct new examples of symmetric algebras.