First of all, note that compounds without -er (Einauge, Zweirad, Dreisatz, Dreiklang, Dreispitz, Vierauge, Fünfkampf) are more frequent than those with -er. I therefore assume compounds with -er have a narrower meaning.
This is borne out by the following observation. There are two ways to form number nouns: By way of zero derivation (feminine, die Eins, Zwei, Drei, etc.) or by adding -er (masculine, der Ein(s)er, Zweier, Dreier, etc.). The latter are more restricted than the former: They are mostly used to refer to grades (ein Fünfer/eine Fünf in Mathe) and money (Ich hab nur noch 'n Fünfer, einen Fünfeuroschein).
The pattern I can discern is that in compounds with -er, the number seems to refer to the internal structure of the object denoted by the head noun.
Zweierzimmer Zimmer mit zwei Betten (Swiss German)
Dreierbeziehung Beziehung zwischen drei Personen
Viererabteil Abteil für vier Personen
Viererpakt Pakt zwischen vier Parteien
Note that these compounds can sometimes be unpacked by adding a noun for the number to count. Then -er vanishes.
Zweibettzimmer, Vierpersonenabteil, Viermächtepakt
The examples you found look like mistakes to me; it should be Vierergruppe in all instances; this is the term Klein used. However, there seem to be some rare (!) cases of missing -er that look genuine. This dissertation from 1908 has Eins-, Zwei-, Drei-, Viergruppe. This could be related to the -er seen as a Fugenlaut and being stigmatized (there once was an attempt to change verfassungsgebend in the German constitution to verfassunggebend).
Having read the other answers to this question, I see that it is highly unclear what kind of -er this is: a nominal suffix, a genitive suffix or a Fugenlaut.
