The Jordan-Hölder Theorem asserts that if a group $G$ has two composition series $$ \{1\}=H_0\lhd H_1\lhd\dots\lhd H_n=G $$ and $$ \{1\}=K_0\lhd K_1\lhd\dots\lhd K_m=G \, , $$ then they are essentially the same: $n=m$, and the composition factors $H_1/H_0,\dots,H_n/H_{n-1}$ are the same (up to isomorphism and permutation) as $K_1/K_0,\dots,K_n/K_{n-1}$.
Because of the phrase "up to isomorphism and permutation", it appears that there are examples of where it is not the case that $H_{i+1}/H_i\cong K_{i+1}/K_i$ for each $i$; rather, there is a permutation $\sigma\in S_n$ such that $\sigma\neq\operatorname{id}$ and $H_{\sigma(i+1)}/H_{\sigma(i)}\cong K_{i+1}/K_i$. Is there an example of this?
The smallest group $G$ for which permuting the composition factors is necessary is $\mathbb Z/6\mathbb Z$. Note that $$ \{\bar0\}\lhd \{\bar0,\bar3\}\lhd\mathbb Z/6\mathbb Z $$ and $$ \{\bar0\}\lhd \{\bar0,\bar2,\bar4\}\lhd \mathbb Z/6\mathbb Z \, . $$ In both cases, the composition factors are, up to isomorphism, $\mathbb Z/2\mathbb Z$ and $\mathbb Z/3\mathbb Z$, but the order in which they appear is different.