Cyclic Frattini quotient implies cyclic

Statement
Let $$G$$ be a group such that the following two conditions:


 * 1) The Frattini subgroup $$\Phi(G)$$ is a finitely generated group (note that this is automatically satisfied if $$G$$ is a finite group)
 * 2) The Frattini quotient, viz., the quotient by the Frattini subgroup, is a cyclic group

Then, $$G$$ is a cyclic group.

Facts used

 * Frattini subgroup is finitely generated implies subset is generating set iff image in Frattini quotient is

Proof
The proof is more or less direct from the above stated fact.