The background problem is exercise 22 in section 6.1 Abstract Algebra by Dummit & Foote:
When $G$ is a finite group prove that if $N\unlhd{G}$ then $\Phi(N)\leq{\Phi(G)}$. Give an explicit example where this containment does not hold if N is not normal in $G$. ($\Phi(G)$ is the Frattini group subgroup of $G$, i.e. the intersection of all maximal subgroups of $G$.)
The proof of the containment when $N$ is normal in G is:
Suppose that $M$ is any maximal subgroup of $G$, if $\Phi(N)\nleq M$, then $N \nleq M$, since $N\unlhd G$ and $\Phi(N) $ is characteristic in $N$, $\Phi(N) \unlhd G$, so $\Phi(N)M$ is a subgroup and $M<\Phi(N)M$, so $\Phi(N)M=G$. Then we have $$N=G\cap N = \Phi(N)M \cap N=\Phi(N)(M\cap N)$$since $M\cap N<N$, consider the maximal group $H$ containing $M\cap{N}$ in $N$, we have $\Phi(N)\leq H$ and $M\cap N\leq H$ so $$N=\Phi(N)(M\cap N)\leq H <N$$ a contradiction, thus we proved that every maximal subgroup of $G$ contains $\Phi(N)$, so we get $$\Phi(N)\leq\Phi(G)$$
But I have no idea about how to construct an example where $N$ is not normal in $G$ and the containment doesn't hold. I have listed some common groups such as $D_{2n},S_n,Q_8,Z_n$ but in many cases the work of computing the Frattini group subgroup is not straightforward, especially for $S_n$, and usually this will take lots of time to enumerate such groups one by one, and I can't get much information from the proof of the former statement, my question is:
1. How to construct a counterexample for this problem?
2. How to work more efficiently when constructing examples in group theory or abstract algebra? This question may sound somewhat abstract and I know that there is no panacea for math discovery, however I think that in the old and well developed math area, namely the basic abstract algebra there may be some experience that could minimize unnecessary work of beginners.
3. Is there any advice for improving the question?
Any help would be appreciated!
This is perhaps an answer to Question 2 with reference to Question 1, but without answering Question 1 explicitly.
An obvious (I hope) idea of where to start is to look for a group $G$ with $\Phi(G)=1$ containing a subgroup $N$ with $\Phi(N) \ne 1$.
An equally obvious class of examples $G$ with $\Phi(G)=1$ is the class of (finite) simple groups.
Now every finite group $N$ embeds in a simple group $A_n$ for some $n \ge 5$, so we just need to find any finite group $N$ with $\Phi(N) \ne 1$ and then embed it into a suitable $A_n$.