well the first line of thinking says
"Event A is very unlikely. But if we suppose that there exists an entity Z whose purpose is to make event A happen, then the likelihood of event A happening is higher"
Which is true, given that entity Z actually has the possibility to influence the happening of A or not. Note that this is nothing more than a truism.
"How does it make sense to infer the existence of a group from a sequence of events?"
Well now the question becomes
"Suppose we have a series of events A,B,C,D. All of them unlikely. They all happen though; this is very weird. It can be explained by assuming that there exists and entity Z that is increasing the probability of A,B,C,D of happening"
Now this is one line of thinking. But this "probabilistic approach" fails for several reasons:
First, nobody assures you that the entity Z is "likely" to exists. I mean if for explaining an event that only happens one time in a thousand you claim that there exists and organization whose "probability to exists" is one in a billion, you didn't gain much, did you?
Second, who decides what events needs explaining and what not? This is a delicate point and actually makes most of this kind of arguments fail. Go to your favourite cinema, and look at plaque of a car; let's say it's AB292ZXZ. Amazing isn't it? From all the billions of combinations possible, that is the one that appears right in front of you! Surely can't be just chance, can it? There must exist an entity to explain this very unlikely event!
The point of the last example is that the last event is unlikely if you say "Today I'll go at a random cinema and I'll see the plaque number AB292ZXZ". It's not unlikely if you just look at a random plaque at the cinema and wonder how incredible it is that you observed that one. There is a big difference between a priori prediction and a posteriori explanation.
The conspiracy theories usually try to explain the world a posteriori; they select some events in a purely arbitrary manner and make up imaginative explanations that "connect all the dots" and explain everything. Well this amounts to nothing, as the plaque example shows; you have to make some accurate predictions. At this they usually fail.
To sum up, you could try to explain events by making up secretive organizations that influence our world, but as long as you can't make meaningful predictions about the future it all amounts to nothing.