Father Robert Southwell the spiritual advisor to Henry Wriothesley and also was a gifted poet whose work influenced Shakespeare-there is even some evidence that Southwell and Shakespeare knew each other. Southwell was arrested , cruelly tortured and ultimately executed by Elizabeth's orders. The fact that a jesuit was an influence on Shakespeare's writing is an indication that Shakespeare himself may have had Catholic leanings. His execution would not have endeared Wriothesley to Elizabeth.
It remains unclear who betrayed Southwell to the authorities but I suspect that it was the scholar John Florio .
John Florio was a tutor for Wriothesley put in place by William Cecil. Florio had very strong anti-Catholic sentiments and later in his life he boasted that his actions had help to capture a number of Jesuits.
Florio was closely linked to Wriothesley so it may well have come to his attention that Southwell was offering Wriothesley spiritual guidance.
Southwell's execution was not popular and occurred on 21 feb 1595 . Charles Blount was present and attempted to stop the execution. It is likely that he was being influenced by Penelope who had strong Catholic sympathies.
Penelope Rich had strong leanings towards Catholicism and was almost converted by Father John Gerrard.
Both Penelope and Robert Devereux also had good reason to dislike Queen Elizabeth because their family was forced by royal decree to finance Elizabeth's war in Ireland - this effectively bankrupted the Devereux family. Also Elizabeth had used The Devereux's family seat Chartley castle and converted it into a prison for Queen Mary of Scots where she was held until her execution.
I think that despite outward appearances the full Essex circle were not royalists and that their anti royalist sentiments ultimately manifested themselves in the Essex rebellion.