Jesuits have been making enemies since their beginning. They have had more enemies (and more dangerous enemies) inside the Roman Catholic communion than outside of it.
Part of it has to do with their methods, which have offended different groups within the church over time, and part of it has been just plain politics. It is oddly ironic that this same order which sent dozens of men to holy martyrdom in England found itself banned as well from Catholic Spain and Portugal (I think France as well, not sure about that). The king of Spain rounded up the Spanish Jesuits (thousands of men) and sent them all by sea to the Papal States, which purportedly refused to let them land.
A Pope dissolved the order. It survived by technicality in Protestant and Orthodox countries which did not ordinarily allow Papal decrees to be published. So even though the Jesuits knew they had been dissolved they continued to function openly as Jesuits, as if nothing happened, under the protection of non-Catholic governments. Some Jesuits from western Europe fled to the areas (Prussian German territories and the Empire of Russia) where the decree had not been published. This kind of rationale seems to exemplify the sort of argumentation typically known as “jesuitical”.
Bishop John Carroll, first bishop of Baltimore, was a Jesuit teaching in Europe until the dissolution. He then returned to North America. and engaged in parish work. It is unclear to me whether the Jesuits in British North America had considered themselves dissolved by the Pope at this time, but a strong hint comes from the founding of Georgetown University by John Carroll in 1789 as a Jesuit school. The order had been disbanded by Pope Clement XIV from 1773 and technically did not exist until resurrected in 1814 by Pope Pius VII.
Eventually reconstituted they proved very reliable to the Papacy. They were very heavily involved in the organization and management of Vatican Council I under Pope Pius IX.
The Jesuits had some remarkable episodes in their history. The can be credited with practically saving Poland on their own from Calvinism, which was spreading around among the upper classes. By educating the younger generations of nobility they managed to bring them back to Roman Catholicism. One imagines young enthusiastic college aged Catholic nobles with hard-bitten old Calvinist grandparents.
On the other hand the Jesuits managed to take over the nascent Russian Catholic movement and nearly killed it.
At one time members of the Society of Jesus could not be bishops. They would necessarily resign from the order if selected for that post. (They worked so independently of the local hierarchy in some parts of the world, such as the Reducciones in S. America, that they were facetiously criticized as a type of “Catholic Presbyterian”.)
I do not think that is the case any more, I feel almost certain that I have read of bishops who identify themselves as Jesuits these days.
In the reign of Pope John Paul II they were definitely not the influential group they had once been, the late Pope was much more comfortable with Opus Dei advice and services. This may be (as some say) because of a general trend on the Jesuit’s part to liberalism in modern times. I don’t have any idea where they stand today with Pope Benedict. It seems to me that the Vatican functioning apparatus may still be prone to factionalism, but more likely based upon personalities and probably not along lines of religious orders as has happened in the past.
I think that the Society of Jesus’ most notable function today is the running of schools, most of which were founded long ago. Some of these are very prestigious institutions.
Most of the current conspiracy theories are just silly beyond words. It’s as if Jack Chick’s imagination is running wild.