I'm still trying to learn projective representation theory,  so please bear with me. Thanks to YCor and David Speyer for pointing out the issues with my previous formulation of this question. Hopefully this is better. 


Below my field will always be $\mathbb{C}$. 


Is there a classification of finite groups G with the property that there exist $c\in H\_{2}(G,\mathbb{Z})$, such that there is a unique irreducible linear representation of the central extension of G associated to c that descends to an irreducible projective representation of G? 


The trivial group trivially has this property. 


The first nontrivial example if this behavior is the klein four group. Specifically, $D\_8$ is the central extension of klein four associated to the unique contrivial homology class. Moreover, there is exactly one 2-dimension irreducible representation that descends to an irreducible projective representation of klein four. 


As a non-example take a finite nontrivial cyclic group G of order n. Then the Schur multiplier is trivial so there are exactly n inequivalent linear representations G that descend the (trivial) Central extension of G. 



Groups with one simple projective representation
Previous answer was broken, here is what I can salvage.


To be clear, I am taking a projective representation to be a map $G \to PGL(V)$ with $\dim V \geq 2$. I call it irreducible if the corresponding representation $\tilde{G} \to GL(V)$, where $\tilde{G}$ is a central extension of $G$, is irreducible. I call two projective representations $G \to PGL(V)$ and $G \to PGL(W)$ isomorphic if I can choose an isomorphism $V \cong W$ to make them coincide. The goal is to classify $G$ such that $G$ only has one irreducible projective representation up to isomorphism. 


Let $A=[G,G]$ and $B = G^{ab}$, so we have $1 \to A \to G \to B \to 1$. I can show


* $A$ is elementary abelian (possibly trivial).
* The conjugacy action of $B$ on $A \setminus \{ e \}$ is transitive.
* $B$ is either of the form $C\_m$ or $C\_{2m} \times C\_2$. (Here $C\_k$ is the cyclic group of order $k$.)


This is a pretty limited list; I am not sure which groups on this list have the stated property.


So, let's prove this. Define $A=[G,G]$ and $B = G^{ab}$ and suppose $G$ has the claimed property.


**Lemma** All non-identity elements of $A$ are conjugate in $G$. (The old version of this answer said conjugate in $A$; that's false.) 


**Proof** Suppose for the sake of contradiction that $A$ has $\geq 3$ conjugacy classes in $G$. Since the character table is invertible, there are two characters, $\chi\_1$ and $\chi\_2$ with $\chi\_i$ coming from $\rho\_i : G \to GL(V\_i)$ which are linearly independent of each other and of the trivial character. Any one dimensional rep of $G$ is trivial on $A$, so $\dim V\_1$ and $\dim V\_2 \geq 2$. 


By assumption, the projective representations $\bar{\rho}\_i : G \to PGL(V\_i)$ are isomorphic, choose an isomorphism $V\_1 \cong V\_2$ such that $\bar{\rho}\_1 = \bar{\rho}\_2$. Then $\rho\_1(g) = \chi(g) \rho\_2(g)$ for some $\chi: G \to \mathbb{C}^{\ast}$. It is easy to see that $\chi$ is a (one-dimensional) character of $G$. But then $\chi$ must be trivial on $A$, so $\chi\_1 = \chi\_2$ on $A$, contradicting our linear independence. $\square$


So the conjugacy action of $G$ on $A$ is transitive on $A \setminus \{ e \}$. By a standard lemma, this implies $A$ is elementary abelian. We've proved the first two bullet points.


Now, we study the structure of the abelian group $B$. Any irreducible projective representation of $B$ is an irreducible projective representation of $G$, so we deduce that $B$ has at most one irreducible
projective representation (none if $B$ is cyclic.) 


Now, if $m>2$, then $C\_m^2$ has multiple irreducible projective representations corresponding to different representations of the Heisenberg group of order $m^3$. (We can see this from the above argument: the commutator subgroup of the Heisenberg group is central, so the conjugacy action on the commutator subgroup is trivial and can't be transitive.) So $B$ must not have $C\_m^2$ as a quotient for $m>2$. Also, $C\_2^3$ has multiple irreducible representations, by projecting onto $C\_2^2$ in different ways. Looking at the classification of finite abelian groups, this limits us to $C\_m$ or $C\_2 \times C\_{2m}$.


I have not figured out which groups on the above list actually have the specified property. 




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I now have the list of candidates down to the following:


$C\_m$ -- this has no irreducible projective representations, so you can decide whether to count it or not.


$C\_{2m} \times C\_2$. The only irreducible projective representation of this is through the quotient $C\_2 \times C\_2$, so it works.


$C\_p^k \rtimes C\_{(p^k-1)t}$, where the action is by a cyclic subgroup of $GL(\mathbb{F}\_p)$ of order $p^k-1$. I have not yet figured out whether or not this works.


$G \times C\_s$ where $s$ is odd and $G$ is a non-abelian central extension of the form $1 \to C\_2 \to G \to C\_{2^k} \times C\_{2} \to 1$. I can show that all projective representations of this factor through the quotient $G$, but I haven't figured out for which $G$ there is a unique irreducible projective representation. 


I'll try to add proofs of what I have later today.


