Perhaps what you are seeking are "classification theorems", namely theorems which have the following format: If $G,H$ are two finite groups having property $P$ then $G$ is isomorphic to $H$ if and only if $G$ and $H$ have the same properties $Q_1,...,Q_n$.
For example, the classification of finite cyclic groups says that if $G$ and $H$ are two finite cyclic groups then $G$ is isomorphic to $H$ if and only if $G$ and $H$ have the same order. So if you know the two groups are cyclic, all you have to do is count how many elements they have to determine whether they are isomorphic.
The next step up is the classification of finite abelian groups. First one proves a theorem: for every finite abelian group $G$ there exists a direct sum decomposition of the form $G = G_1 \oplus G_2 \oplus \cdots G_K$ such that each $G_i$ is a finite cyclic group of some order $d_i$, and such that the sequence of group orders is linearly ordered by divisibility: $d_1$ divides $d_2$ which divides $d_3$ which divides... which divides $d_K$. There's a word for this sequence $d_i$ which I don't know, so I'll just temporarily call it the "divisor sequence" of the group $G$. The classification theorem says that if $G$ and $H$ are finite abelian groups then $G$ is isomorphic to $H$ if and only if they have the exact same divisor sequence.
There are a lot of examples of theorems like this, of more and more power applying to more and more general classes of finite groups. They get harder and harder to prove, as the class becomes more and more general.
Keep in mind, though, that there is an important reality hidden in the proofs: when you trace through the proof of existence of an isomorphism between two finite groups, you will almost always find the construction of an isomorphism. (In finite group theory, I doubt there's a single exception to this rule. In infinite group theory, there are probably many exceptions in which the proof of existence of an isomorphism leads instead to some existence axiom such as the axiom of choice).