Let $X$ be a smooth complex projective curve and let $x\in X$ be a point. We compute the automorphism group of the moduli space of framed vector bundles on $X$ with a framing over $x$. It is shown that this group is generated by pullbacks using automorphisms of the curve $X$ that fix the marked point $x$, tensorization with certain line bundles over $X$ and the action of $\operatorname{PGL}_r(\mathbb{C})$ by composition with the framing. A Torelli theorem for framed bundles with a generic stability parameter is also proved. Joint work with Indranil Biswas.