Motivated by a question of Ghys, we talk about cohomological obstructions to extending group actions on the boundary $\partial M$ of a $3$-manifold to a $C^0$-action on $M$. Among other results, we show that for a $3$-manifold $M$, the $\mathbb{S}^1 \times \mathbb{S}^1$ action on the boundary does not extend to a $C^0$-action of $\mathbb{S}^1 \times \mathbb{S}^1$ as a discrete group on $M$, except in the trivial case $M \cong \mathbb{D}^2 \times \mathbb{S}^1$. Using additional techniques from 3-manifold topology, homotopy theory, and low-dimensional dynamics, we find group actions on a torus and a sphere that are not nullbordant, i.e. they admit no extension to an action by diffeomorphisms on any manifold $M$ with $\partial M \cong \mathbb{T}^2$ or $\mathbb{S}^2$. This is a joint work with K. Mann.