On invariant subalgebras when the ISR property fails

We classify all G-invariant von Neumann subalgebras in L(G) for G=Z2⋊SL2(Z). This is the first result on classifying G-invariant von Neumann subalgebras in L(G) for icc groups G without the invariant von Neumann subalgebras rigidity property (ISR property for short) as introduced in Amrutam--Jiang's work. As a corollary, we show that L(Z2⋊{±I2}) is the unique maximal Haagerup G-invariant von Neumann subalgebra in L(G), where I2 denotes the identity matrix in SL2(Z).