  We discuss first-principles topological electronic structure of
noncentrosymmetric SrSi$_2$ materials class based on the hybrid
exchange-correlation functional. Topological phase diagram of SrSi$_2$ is
mapped out as a function of the lattice constant with focus on the semimetal
order. A tunable double-Weyl Fermion state in Sr$_{1-x}$Ca$_{x}$Si$_2$ and
Sr$_{1-x}$Ba$_{x}$Si$_2$ alloys is identified. Ca doping in SrSi$_2$ is shown
to yield a double-Weyl semimetal with a large Fermi arc length, while Ba doping
leads to a transition from the topological semimetal to a gapped insulator
state. Our study indicates that SrSi$_2$ materials family could provide an
interesting platform for accessing the unique topological properties of Weyl
semimetals.
