For solvable groups without Frattini chief factors, this is equivalent to each of the following (individually):

* having a unique chief series,
* every quotient group having a faithful primitive permutation action,
* the upper Fitting series being a chief series
* the lower Fitting series being a chief series

This is shown in:
>Hawkes, Trevor O.
"Two applications of twisted wreath products to finite soluble groups."
Trans. Amer. Math. Soc. 214 (1975), 325–335.
MR<a href="http://www.ams.org/mathscinet-getitem?mr=379657">379657</a>
DOI:<a href="http://dx.doi.org/10.2307/1997110">10.2307/1997110</a>

You might also be interested in the safari for <a href="https://math.stackexchange.com/questions/14919/zebra-groups-and-counting-stripes">zebra groups</a>.

However, there are solvable groups with Frattini factors whose normal subgroups form a chain: SL(2,3) for example.