Does there exist a complete classification of all fiber bundles $\Bbb S^k\longrightarrow\Bbb S^d\overset{\smash{\pi}}\longrightarrow B$, that is, fibrations of $\smash{\Bbb S^d}$ with each fiber homeomorphic to $\smash{\Bbb S^k}$ for some fixed $k\le d$.

Wikipedia contains a list of some real/complex/quaternionic/octonionic fibrations. In other words: is this list complete?

I am then interested, which of the base spaces $B$ that appear in above classification admit a topological/Lie group structure (compatible with its present topology).