= Solution
The continuous internal symmetry is local $U(1)$ <gauge invariance>. With the source's <gauge covariant derivative> convention it acts as
$$
\boxed{\varphi\mapsto e^{i\alpha(x)}\varphi,\qquad
A_\mu\mapsto A_\mu+\frac1e\partial_\mu\alpha,}
$$
so $D_\mu\varphi$ transforms like $\varphi$ and $F_{\mu\nu}$ is unchanged. Constant phases are its global subgroup, not an additional independent internal rotation group.
In flat four-dimensional space-time the continuous external symmetry is the <Poincaré group>: four translations and six <Lorentz transformations>. Noether's theorem associates translations with conserved energy and momentum, spatial rotations with <angular momentum>, and boosts with the relativistic center-of-energy/boost charges. In tensor notation the latter two are the six components of the conserved total angular-momentum tensor $M^{\mu\nu}$, including field spin contributions where appropriate. The nonzero dimensionful parameter $\mu^2$ prevents exact scale invariance. Discrete transformations, such as parity, are outside the requested continuous-symmetry list.
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