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. MR379657 DOI:10.2307/1997110

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

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. MR379657 DOI:10.2307/1997110

You might also be interested in the safari for zebra groups.

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