Let $\pi: M \to N$ be a submersion and $\mathcal{H}$ a connection on $M$. For a piecewise $C^k$-path $c: [0,1] \to N$ in $N$ let $\tau_c: \pi^{-1}(c(0)) \to \pi^{-1}(c(1))$ be the parallel transport along $c$.
There are holonomy groups $\text{Hol}_k(x)$ defined as the group of all parallel transport diffeomorphisms $\tau_c$, where $c$ is a piecewise $C^k$-loop at $x\in N$, and holonomy groups $\text{Hol}_k^{'}(x)$ defined as the group of all parallel transport diffeomorphisms $\tau_c$, where $c$ is a $C^k$-loop at $x\in N$ (not piecewise).
If $(M,\pi,N)$ is a $G$-principal bundle then $\text{Hol}_k(x) = \text{Hol}_l(x)$ for all $k,l = 1,...,\infty$. Does this also hold for general submersions? And does $\text{Hol}_k^{'}(x) = \text{Hol}_k(x)$ hold?