Such groups are solvable. This has been solved by Howlett and Isaacs himself, in

Howlett, Robert B.; Isaacs, I. Martin, On groups of central type, Math. Z. 179, 555-569 (1982). ZBL0511.20002.

The proof uses the classification of finite simple groups.

Such groups are solvable. This has been solved by Howlett and Isaacs himself, in

Howlett, Robert B.; Isaacs, I. Martin, On groups of central type, Math. Z. 179, 555-569 (1982). MR652860 ZBL0511.20002.

The proof uses the classification of finite simple groups.