  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.
