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The disciplinary committee in the HRD ministry has suggested
that the arrest of an employee even for single day or payment of fine in lieu of the same for the commitment of criminal offence should be grounds for dismissing them. (A). that the arrest of an employee even for single day or payment of fine in lieu of the same, for the commitment of criminal offence, should be grounds for dismissing them.
(B) the arrest of an employee even for single day or payment of fine in lieu of the same for the commitment of criminal offence should be grounds for dismissing him.
(C) that the arrest of an employee even for single day or payment of fine in lieu of the same for the commitment of criminal offence, be grounds for dismissing him.
(D) that an employee’s arrest even for single day or payment of fine in lieu of the same for the commitment of criminal offence, should be considered for dismissing him.
(E) that for the commitment of criminal offence, an employee be dismissed because of his arrest even for single day or payment of fine in lieu of the same
Howdy! Happy to help sort through this one.

First thing I noticed when reading was the verb in the part of the sentence that we can't change—"has suggested." Good to make note of. Also, something that jumped out immediately was the idiom "suggested that." From there we can eliminate (B). Sweet!
Not let's dig deeper. Reading over each option, I found (E) to be quite a mess. It reorganizes the sentence to such a degree that it's original meaning seems lost. Let's push this one aside and look at our other options.
With our three remaining answer choices we now have to focus our attention on the final part of the sentence. One question is "Do we need 'should'?" And another question is "Do we need 'be considered for'?" Well this second question we can answer quickly. This is an awkward and unnecessary construction. So we can toss this answer choice out and now we are left with two choices left.
Do we need 'should'? The answer is no. Including "should" is redundant since we already know that the committee is suggesting something. As always, we need to find the most precise and concise formulation of an idea in SC questions, and in this case, "should" only repeats an idea that occurs at the beginning of the sentence. So we can eliminate the answer choice with "should," chose the correct answer, and move on to other questions!

Hope this helps!