The center is remembering. The pa'al לִזְכֹּר is to remember; the hif'il לְהַזְכִּיר is to remind (cause to remember) and also to mention; the nif'al לְהִזָּכֵר is to recall, to come to mind. The nouns cluster the same idea: a private faculty (זִכָּרוֹן, memory), a public act (הַזְכָּרָה, a mention; אַזְכָּרָה, a commemoration), and a practical object (תִּזְכֹּרֶת, a reminder). One word, מַזְכִּיר, is the person whose job is to keep things on the record: a secretary. A famous homonym also sits on this root: זָכָר, "male", which Hebrew tradition links to the one who "is mentioned / named". Every word here is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
Watch the three voices on one idea: pa'al לִזְכֹּר is the plain act (I remember), nif'al לְהִזָּכֵר is the passive / spontaneous version (it comes back to me, it is remembered), and hif'il לְהַזְכִּיר is the causative (I make you remember, i.e. remind, or I bring a thing up, i.e. mention). The passive pu'al and huf'al are not lexicalized for this root, which is normal: not every root fills all seven binyanim.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.