Here is the technical explanation:
What we're looking at here is "aspect," or how events are distributed through time. When we think of *tense, we usually just look at past, present, and future. In reality, because of aspect we are able to indicate whether an action is a single occurrence, a repeated or habituated occurrence, continuing or terminated, etc.
Examples:
I eat a sandwich for lunch.
I am eating a sandwich for lunch.
I have eaten a sandwich for lunch.
These are all technically "present" tense, but the meaning is different between each sentence because the aspect is different.
*You will hear verbal aspect described as "tense," but that is not really correct since you can combine them, and it doesn't make sense to have different tenses occurring in the same clause for the same verb. That is, a verb can't be both present and past tense at the same time. It can however, be continuing or completed.
Perfect aspect: expresses the
completion of an action or event with respect to some reference point in time.
The action of the verb is complete at the time of the reference point.
Construction: "have" + "-ed" or "-en" (e.g. have walk
ed, have eat
en)
Past Perfect: has a reference point that is prior to the present moment.
She had walked the dog.
Present Perfect: the reference point is in the present (i.e., the time when the sentence is spoken).
She has walked the dog.
Future Perfect: has a reference point in the future.
She will have walked the dog.
Progressive aspect: also specifies a reference point in the past, present or future, but the action of the verb is
in progress at the time of the reference. In other words, the action begins before the reference point and continues beyond the reference point for an unspecified amount of time. (i.e., began in the past and continued).
Construction: "to be" + "-ing" (e.g., I am walk
ing)
Past Progressive: the reference point and moment of continuation are in the past, prior to the moment of speaking.
She was walking the dog.
Present Progressive: the action continues through the present moment.
She is walking the dog.
Future Progressive: the action will begin and continue into the future.
She will be walking the dog.
Perfect Progressive aspect: basically combines the other two, indicating continuing action that was completed at the time of the reference point. (i.e., began in the past, continued, then completed).
Construction: "have" + "been" + "-ing" (e.g, I
have been walk
ing)
Past Perfect Progressive: the reference point, moment of continuation and completion are in the past, prior to the moment of speaking.
She had been walking the dog.
Present Perfect Progressive: the action began, continued and completed up to the present moment.
She has been walking the dog.
Future Perfect Progressive: the action will begin, continue and complete at some point in the future.
She will have been walking the dog.
So on to your sample sentence:
"The day after we sent that invitation, we heard that X group has decided to expand their Friday group to two times a month, just as they used to have it."
"We heard that X group has decided...." This event (the deciding) technically occurred before the invitation was sent out, so it was completed in the past. -->Hmm....sounds a lot like past perfect.
It should read "we heard that X group
had decided..."
I'm not sure of a good way to explain this, but it is only past perfect because of the way it's tied to the first clause. "The day after we sent that invitation" basically established a timeline of events and the reference points for the rest of the sentence.
If you separate the clauses into two sentences (severing the tie, so to speak) it doesn't sound as awkward:
"We sent out the invitations. Then we heard that X group has decided to expand...."
Another option: "After sending out the invitations, we heard that X group has decided to expand." By putting "sending" in the sentence, it brings the first clause into the present, which then matches the reference point in the second clause.
Ultimately plain "decided" would have been better than "has decided," because it is simple past and would have matched the preceding clause (which was also past tense).
I hope this made sense (it's very late, and I'm obviously an insomniac
)