The adverbial ending -ically can be awkward to handle: in general, we have to remember to write the -al- in the middle, but not to pronounce it.
English has hundreds of adjectives ending -ic. The most common include artistic, athletic, authentic, basic, classic, cosmetic, domestic, dramatic, dynamic, economic, electric, electronic, exotic, fantastic, historic, Islamic, optimistic, organic, public, realistic, romantic, scientific, specific, static, systematic, toxic, tragic.
English also has a smaller but substantial number of adjectives ending -ical. The most common include anatomical, biological, classical, critical, economical, electrical, historical, identical, medical, musical, physical, political, practical, psychological, radical, technical, topical, tropical, typical.
(There are quite a few pairs, eg classic and classical, economic and economical, historic and historical, with somewhat different meanings or uses.)
In writing, the general rule is that both sets are turned into adverbs ending -ically (though there’s one major exception: publicly). In pronunciation, however, these are all generally treated as if they were -icly. Here are native speakers (mainly British) saying basically, typically, historically and systematically:
Of course, native speakers tend to write the way they speak, so they often leave out the -al- in writing. On the site Tripadvisor, this review has the headline “Basicly the cheapest place to stay”, and this review describes a guesthouse as “historicly interesting”. Non-natives, on the other hand, tend to speak the way they write, so they often pronounce the -al- in -ically, thereby making themselves sound foreign.
Some dictionaries tell you that it’s equally possible to pronounce -ically as /ɪkəli/, but in fact this is quite rare: learners and users who want to sound native should avoid the /ə/.
Just out of curiosity, what is the situation for -entally? Are accidentally and evidently (when third syllable is stressed) pronounced the way they are spelled, or do they conflate, with or without a syllable for the -tl-/-tal- sequence?
Good question. The endings of accidentally and evidently are unlikely to conflate in BrE, where evidently is generally stressed on the first syllable.
Stressing the third syllable is more common in AmE than in BrE. In AmE the endings of the two words may conflate, but often don’t: the option exists to pronounce accidentally as five syllables.
It’s my impression that (in both BrE and AmE) the reduction of -entally to -ently is only an option in accidentally and incidentally: I don’t think you’ll hear it in eg experimentally or environmentally or fundamentally.