From BiblePortal Wikipedia

Vine's Expository Dictionary of NT Words [1]

1: Δοκιμάζω (Strong'S #1381 — Verb — dokimazo — dok-im-ad'-zo )

"to prove with a view to approving," is twice translated by the verb "to allow" in the AV; the RV corrects to "approveth" in  Romans 14:22 , and "have been approved,"  1—Thessalonians 2:4 , of being qualified to be entrusted with the Gospel; in  Romans 1:28 , with the negative, the RV has "refused," for AV, "did not like." See Approve.

2: Γινώσκω (Strong'S #1097 — Verb — ginosko — ghin-oce'-ko )

"to know," is rendered "allow" in  Romans 7:15 (AV); the RV has "that which I do I know not;" i.e., "I do not recognize, as a thing for which I am responible." See Aware , Can , Feel , Know , Perceive , Resolve , Speak , Sure , Understand.

3: Συνευδοκέω (Strong'S #4909 — Verb — suneudokeo — soon-yoo-dok-eh'-o )

"to consent or fully approve" (sun, "with," eu, "well," dokeo, "to think"), is translated "allow" in  Luke 11:48; "was consenting" in  Acts 8:1;  22:20 . See Consent.

4: Προσδέχομαι (Strong'S #4327 — Verb — prosdechomai — pros-dekh'-om-ahee )

mistranslated "allow" in  Acts 24:15 , AV, means "to wait for," in contrast to rejection, there said of entertaining a hope; hence the RV, "look for." See Accept , A, No. 3.

Webster's Dictionary [2]

(1): (v. t.) To own or acknowledge; to accept as true; to concede; to accede to an opinion; as, to allow a right; to allow a claim; to allow the truth of a proposition.

(2): (v. t.) To grant license to; to permit; to consent to; as, to allow a son to be absent.

(3): (v. t.) To grant (something) as a deduction or an addition; esp. to abate or deduct; as, to allow a sum for leakage.

(4): (v. i.) To admit; to concede; to make allowance or abatement.

(5): (v. t.) To like; to be suited or pleased with.

(6): (v. t.) To sanction; to invest; to intrust.

(7): (v. t.) To grant, give, admit, accord, afford, or yield; to let one have; as, to allow a servant his liberty; to allow a free passage; to allow one day for rest.

(8): (v. t.) To praise; to approve of; hence, to sanction.

King James Dictionary [3]

Allow', L loco, to lay, set, place. See Lay.

1. To grant, give or yield as, to allow a servant his liberty to allow a pension. 2. To admit as, to allow the truth of a proposition to allow a claim. 3. To admit to own or acknowledge as, to allow the right of the President to displace officers. 4. To approve, justify or sanction.

Ye allow the deeds of your fathers.  Luke 11 .  Romans 8 .

5. To afford, or grant as a compensation as, to allow a dollar a day for wages. 6. To abate or deduct as, to allow a sum for tare or leakage. 7. To permit to grant license to as, to allow a son to be absent.

Hastings' Dictionary of the Bible [4]

Allow . To ‘allow’ generally means in AV [Note: Authorized Version.] ‘to approve,’ as   Romans 7:15 ‘that which I do I allow not.’ But in   Acts 24:16 it has the mod. sense, admit .