I've seen there is a similar question, but it looks for something more specific. I'll write what I thought.
I've rapidly seen that every group of order 28 has a unique normal subgroup of order 7. Then exists an element $a$ such that $\langle a \rangle =: H$ is normal.
I have that it has to be 1 2-Sylow or 7 2-Sylow subgroups. If there is only 1, then $G$ is abelian, so $G \cong \mathbb{Z}_4 \times \mathbb{Z}_7$ or $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_7$.
If there are 7 2-Sylow subgroups, I have a harder time finding the possibilities. If there exists an $b \in G$ such that $|\langle b \rangle|=2$ and $\langle a,b \rangle = G$, then, since $H$ is normal, $b ab^{-1}=a^m$ for $m\in\mathbb{N}$, the $ba=a^mb$. Then:
$$a=b^2a=ba^mb=a^{m^2}b^2=a^{m^2}$$
Then $a^{m^2-1}=1$ so, for Lagrange, $7|m^2-1=(m+1)(m-1)$. I can assume $0 <m <7$ so this leaves me with $m=1$ or $m=6$.
If $m=1$, then $ab=ba$ meaning $G$ is abelian, but it's also generated from elements of order 7 and 2, but there are no elements of order 4 and only one element of order 2, so it's not possible.
If $m=6$, since $\langle a,b \rangle=G$, the mapping $a \rightarrow r$ and $b \rightarrow s$ shows me that $G \cong D_{14}$ the dihedral group.
So, I'm left with the case that $\langle a, b \rangle \neq G$. If I'm not mistaken, I have two cases: either exists another element $c \neq b$ with order 2, or I have a $c$ with order 4 such that $\langle a,c \rangle = G$. I think both cases are the same, but don't know how to justify it.
EDIT
In the first case, I've found that $G\cong D_7 \times \mathbb{Z}_2$. I'm not sure why this is the only non-abelian group formed by 3 elements, with order 7, 2 and 2 respectively.
If $c$ has order 4, then, with the same analysis as before, $ca = ac$ or $ca = a^6c$. In the first case, $G$ is abelian so is a case seen before. In the second case, I'm not sure how to continue. I'm not completely sure if it has to be $\mathbb{Z}_7 $⋊$ \mathbb{Z}_4$ for some indirect product, but I'm not sure how to find all functions such that is an indirect product