  Given a continuous, odd, semi-simple $2$-dimensional representation of
$G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$ and a prime
$\ell$ not dividing $Np$, we study the relation between the universal
deformation rings of the corresponding pseudo-representation for the groups
$G_{\mathbb{Q},N\ell p}$ and $G_{\mathbb{Q},Np}$. As a related problem, we
investigate when the universal pseudo-representation arises from an actual
representation over the universal deformation ring. Under some hypotheses, we
prove analogues of theorems of Boston and B\"{o}ckle for the reduced
pseudo-deformation rings. We improve these results when the
pseudo-representation is unobstructed and $p$ does not divide $\ell^2-1$. When
the pseudo-representation is unobstructed and $p$ divides $\ell+1$, we prove
that the universal deformation rings in characteristic $0$ and $p$ of the
pseudo-representation for $G_{\mathbb{Q},N\ell p}$ are not local complete
intersection rings. As an application of our main results, we prove a big
$R=\mathbb{T}$ theorem.
