Maps between spherical group algebras

I will speak about a central question in higher algebra (aka brave new algebra), namely which rings or schemes admit 'higher models', that is lifts to the sphere spectrum. This question is in some sense very classical, but there are many open questions. These questions are closely related to questions about higher versions of prismatic cohomology and delta ring, asked e.g. by Scholze and Lurie. Concretely we will consider the case of group algebras and explain how to understand maps between lifts of group algebras to the sphere spectrum. The results we present are joint with Carmeli and Yuan and on the prismatic side with Antieau and Krause.