The stem of the second perfect is formed by adding α to the reduplicated verb-stem: γέ-γραφ-α
Hom. has several forms unknown to Attic: δέδουπα (δουπ-έ-ω
The second perfect is almost always formed from stems ending in a liquid or a stop consonant, and not from vowel stems.
a. ἀκήκοα (ἀκούω
Verb-stems showing variation between short and long vowels ( cross476) have long vowels in the second perfect (α is thus regularly lengthened). Thus, τήκω (τακ-, τηκ-)
The second perfect has ο, οι when the verb-stem varies between α, ε, ο ( cross478, cross479) or ι, ει, οι ( cross477 a): τρέφ-ω (τρεφ-, τροφ-, τραφ-)
Similarly verbs with the variation υ, ευ, ου ( cross476) should have ου; but this occurs only in Epic εἰλήλουθα ( = Att. ἐλήλυθα); cp. ἐλεύ (θ) -σομαι. Other verbs have ευ, as φεύγω
After Attic reduplication ( cross446) the stem of the second perfect has the weak form; ἀλείφω (ἀλειφ-, ἀλιφ-) anoint ἀλήλιφα.567
Apart from the variations in 563-566 the vowel of the verb-stem remains unchanged: as γέγραφα (γράφω
The meaning of the second perfect may differ from that of the present; as ἐγρήγορα
Aspirated Second Perfects.—In many stems a final π or β changes to φ: a final κ or γ changes to χ. (φ and χ here imitate verb-stems in φ and χ, as τρέφω, ὀρύττω.)
Hom. never aspirates π, β, κ, γ. Thus κεκοπώς = Att. κεκοφώς (κόπ-τ-ω
Most such stems have a short vowel immediately before the final consonant; a long vowel precedes ε.γ. in δείκ-νυ_-μι δέδειχα, κηρύ_ττω (κηρυ_κ-) -κεκηρυ_χα, πτήσσω (πτηκ-) ἔπτηχα. τέτριφα and τέθλιφα show ι in contrast to ι_ in the present (τρί_βω, θλί_βω). στέργω, λάμπω do not aspirate (ἔστοργα, poet. λέλαμπα).571
The following verbs have aspirated second perfects: ἄγω, ἀλλάττω, ἀνοίγω, βλάπτω, δείκνυ_μι, διώκω (rare), θλί_βω, κηρύ_ττω, κλέπτω, κόπτω, λαγχάνω, λαμβάνω, λάπτω, λέγω
Second Perfects of the μι-form.—Some verbs add the endings directly to the reduplicated verb-stem. Such second perfects lack the singular of the indicative.
ἵστημι (στα-, στη-)
Stem Gradation.—Originally the second perfect was inflected throughout without any thematic vowel (cp. the perfect middle), but with stem-gradation: strong forms in the singular, weak forms elsewhere. -α (1 singular) was introduced in part from the aorist and spread to the other persons. Corresponding to the inflection of οἶδα ( cross794) we expect πέποιθα, πέποισθα, πέποιθε, πέπιστον, πέπιθμεν, πέπιστε, πεπίθατι (from πεπιθτι). Traces of this mode of inflection appear in Hom. γεγάτην (from γεγτην, 35 b) γέγαμεν from γέγονα; ἔϊκτον, ἐΐκτην, ἐϊκώς from ἔοικα; ἐπέπιθμεν; μέμαμεν from μέμονα; πέπασθε (for πεπαθτε πεπθτε) from πέπονθα (other examples 704, cross705). So the masc. and neut. participles have the strong forms, the feminine has the weak forms (μεμηκώς, μεμακυῖα as εἰδώς, ἰδυῖα).
Herbert Weir Smyth [n.d.], A Greek Grammar for Colleges; Machine readable text [info] [word count] [Smyth].