$\mathrm{Fibonacci}(n)$ modulo $n$ sequence

We study the behavior of the $\mathrm{Fibonacci}(n)\mod n$ sequence and pay attention to some subsequences: $n$ runs through the set of prime numbers and the cases with $n = qp$, where $p$ runs through the set of prime numbers and $q$ is a fixed natural number. The behavior of the sequence is investigated using the Mathematica system. Some hypotheses are formulated and proved.