What happens to the $SU(2)_L$ group after spontaneous symmetry breaking? The electro-weak symmetry $SU(2)_L\times U(1)_Y$ spontaneously breaks into $U(1)_{EM}$ - and underlies the Higgs mechanism.
What happens to the $SU(2)_L$ symmetry after symmetry breaking? Why are left handed (right handed) leptons and neutrinos still doublets (singlets) of $SU(2)_L$?
 A: Spontaneous Symmetry Breaking means the symmetry remains a symmetry of the Lagrangian, and the Noether current of the symmetry transformation is still conserved, but the formerly conserved charge is mooted due to infrared effects (cf. the Fabri–Picasso Theorem in the link below). This Nambu-Goldstone realization of the symmetry is, nevertheless, a genuine realization of the symmetry.
The symmetry transformations are now nonlinear for the Higgs double--and the Goldstone boson parts of it eaten up by the gauge bosons. An outstanding review, despite the confusing title is Ramond  1999, ISBN-13: 978-0738201160.
The transformations of the fermions, however,  remain unchanged.  Under these modified transformation laws, the lagrangian is still invariant under this SU(2)L.  Certain relations among constants in the lagrangian will now appear very different to those in the unbroken case.
The ill-defined weak charges of these conserved currents, however, suffer degradation, and you must be quite careful in forensically utilizing remnants thereof where you can segregate the effects of the Higgs doublet---a  risky idea, unless you know what you are doing.
