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Lets Suppose there is Conducting plate X and charge Q is given to which means +Q/2 and +Q/2 on each side . Now lets suppose plate Y (uncharged at first) (exactly similar to plate X (area and material wise) )( is brought near to the plate X . Off course redistribution will happen. Q/2 on both sides of plate X and -Q/2 on the inner side and Q/2 on outer side of plate Y such that E is 0 inside the conductor As Explanation provided in the answer for this question

Now my question is suppose Plate Y's outer side is earthed which means -Q/2 charge will flow from earth to plate y thus 0 charge on outside surface of plate Y. Now the system will behave as capacitor (Parallel plate capacitor ).
But the problem is in the redistribution .The whole charge on plate X's outer side Q/2 will flow inside thus making redistribution something like this +Q on the inner side of Plate x and -Q on the inner side of Plate Y. But now if i take an arbitrary point inside Plate X I'm finding E = 0 inside that conductor(plate X){as Electric field direction will be only from plate x to plate y} but not inside an arbitrary point inside Plate Y?Can anyone explain how will redistribution takes place in an earthed plate

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Your final charge distribution on both plates( Q on inner surface of plate X and -Q on inner surface of plate Y) is correct. The electric field inside both the plate is also zero. Since you you assumed it to be a parallel plate capacitor that means you are treating the plates as infinite surface of charge. In this case the electric field at any point due to a single plate will be constant and proportional to charge per unit area on it. thus electric field would be zero everywhere except for the points between the plates.

Electric field lines will be from X to inner surface of Y but will end at inner surface of Y

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