Let $G$ be a finite group and $\pi$ be a subset of primes dividing order of $G$.
Then $G$ is said to be $\pi$-separable if there is a series of subgroups $1<H_1<H_2\cdots < H_n=G$ such that $H_i$ is normal in $H_{i+1}$ and each successive quotient is either $\pi$-group or $\pi$'-group.
(One successive quotient can be $\pi$-group and other can be $\pi$'-group).
Q. If $G$ is $p$-separable for every prime divisor $p$ of $|G|$ then what can be said about $G$?
The finite solvable groups are of this type (mentioned in question), but $S_n$ for $n\ge 5$ are not $2$-separable, am I right?
These are precisely the groups whose non-abelian composition factors are not divisible by $p$ for any $p\in\pi$.
That this is sufficient is easy as a composition series is a series of the above description. That this is necessary is also easy, because any series of the above description can be refined to a composition series and must therefore contain for each somposition factor $N$ of $G$ some $H_i, H_{i+1}$ with $|N|$ dividing $|H_{i+1}/H_i|$.